### Aub c -

A) = 7/9As can be seen, the probability that a black marble is drawn is affected by any previous event where a black or blue marble was drawn without replacement. Thus, if a person wanted to determine the probability of withdrawing a blue and then black marble from the bag:

Probability of drawing a blue and then black marble using the probabilities calculated above:

P(A ∩ B) = P(A) × P(B c B). Take the example of a bag of 10 marbles, 7 of which are black, and 3 of which are blue. Calculate the probability of drawing a black marble if a blue marble has been withdrawn without replacement (the blue marble is removed from the bag, reducing the total number of marbles in the bag):

Probability of drawing a blue marble:

P(A) = 3/10

Probability of drawing a black marble:

P(B) = 7/10

Probability of drawing a black marble given that a blue marble was drawn:

P(B A) = (3/10) × (7/9) = 0.2333

### Union of A and B

In probability, the union of events, **P(A U B)**, essentially involves the condition where any or all of the events being considered occur, shown in the Venn diagram below. Note that **P(A U B)** can also be written as **P(A OR B)**. In this case, the "inclusive OR" is being used. This means that while at least one of the conditions within the union must hold true, all conditions can be simultaneously true. There are two cases for the union of events; the events are either mutually exclusive, or the events are not mutually exclusive. In the case where the events are mutually exclusive, the calculation of the probability is simpler:

A basic example of mutually exclusive events would be the rolling of a dice, where event **A** is the probability that an even number is rolled, and event **B** is the probability that an odd number is rolled. It is clear in this case that the events are mutually exclusive since a number cannot be both even and odd, so **P(A U B)** would be **3/6 + 3/6 = 1**, since a standard dice only has odd and even numbers.

The calculator above computes the other case, where the events **A** and **B** are not mutually exclusive. In this case:

P(A U B) = P(A) + P(B) - P(A ∩ B)

Using the example of rolling dice again, find the probability that an even number or a number that is a multiple of 3 is rolled. Here the set is represented by the 6 values of the dice, written as:

S = {1,2,3,4,5,6} | |

Probability of an even number: | P(A) = {2,4,6} = 3/6 |

Probability of a multiple of 3: | P(B) = {3,6} = 2/6 |

Intersection of A and B: | P(A ∩ B) = {6} = 1/6 |

P(A U B) = 3/6 + 2/6 -1/6 = 2/3 |

### Exclusive OR of A and B

Another possible scenario that the calculator above computes is **P(A XOR B)**, shown in the Venn diagram below. The "Exclusive OR" operation is defined as the event that A or B occurs, but not simultaneously. The equation is as follows:

As an example, imagine it is Halloween, and two buckets of candy are set outside the house, one containing Snickers, and the other containing Reese's. Multiple flashing neon signs are placed around the buckets of candy insisting that each trick-or-treater only takes one Snickers OR Reese's but not both! It is unlikely, however, that every child adheres to the flashing neon signs. Given a probability of Reese's being chosen as **P(A) = 0.65**, or Snickers being chosen with **P(B) = 0.349**, and a **P(unlikely) = 0.001** that a child exercises restraint while considering the detriments of a potential future cavity, calculate the probability that Snickers or Reese's is chosen, but not both:

0.65 + 0.349 - 2 × 0.65 × 0.349 = 0.999 - 0.4537 = 0.5453

Therefore, there is a 54.53% chance that Snickers or Reese's is chosen, but not both.

### Normal Distribution

The normal distribution or Gaussian distribution is a continuous probability distribution that follows the function of:

where **μ** is the mean and **σ ^{2}** is the variance. Note that standard deviation is typically denoted as

**σ**. Also, in the special case where

**μ = 0**and

**σ = 1**, the distribution is referred to as a standard normal distribution. Above, along with the calculator, is a diagram of a typical normal distribution curve.

The normal distribution is often used to describe and approximate any variable that tends to cluster around the mean, for example, the heights of male students in a college, the leaf sizes on a tree, the scores of a test, etc. Use the "Normal Distribution" calculator above to determine the probability of an event with a normal distribution lying between two given values (i.e. **P** in the diagram above); for example, the probability of the height of a male student is between 5 and 6 feet in a college. Finding **P** as shown in the above diagram involves standardizing the two desired values to a z-score by subtracting the given mean and dividing by the standard deviation, as well as using a Z-table to find probabilities for Z. If, for example, it is desired to find the probability that a student at a university has a height between 60 inches and 72 inches tall given a mean of 68 inches tall with a standard deviation of 4 inches, 60 and 72 inches would be standardized as such:

Given **μ** = 68; **σ** = 4

(60 - 68)/4 = -8/4 = -2

(72 - 68)/4 = 4/4 = 1

The graph above illustrates the area of interest in the normal distribution. In order to determine the probability represented by the shaded area of the graph, use the standard normal Z-table provided at the bottom of the page. Note that there are different types of standard normal Z-tables. The table below provides the probability that a statistic is between 0 and Z, where 0 is the mean in the standard normal distribution. There are also Z-tables that provide the probabilities left or right of Z, both of which can be used to calculate the desired probability by subtracting the relevant values.

For this example, to determine the probability of a value between 0 and 2, find 2 in the first column of the table, since this table by definition provides probabilities between the mean (which is 0 in the standard normal distribution) and the number of choices, in this case, 2. Note that since the value in question is 2.0, the table is read by lining up the 2 row with the 0 column, and reading the value therein. If, instead, the value in question were 2.11, the 2.1 row would be matched with the 0.01 column and the value would be 0.48257. Also, note that even though the actual value of interest is -2 on the graph, the table only provides positive values. Since the normal distribution is symmetrical, only the displacement is important, and a displacement of 0 to -2 or 0 to 2 is the same, and will have the same area under the curve. Thus, the probability of a value falling between 0 and 2 is 0.47725 , while a value between 0 and 1 has a probability of 0.34134. Since the desired area is between -2 and 1, the probabilities are added to yield 0.81859, or approximately 81.859%. Returning to the example, this means that there is an 81.859% chance in this case that a male student at the given university has a height between 60 and 72 inches.

The calculator also provides a table of confidence intervals for various confidence levels. Refer to the Sample Size Calculator for Proportions for a more detailed explanation of confidence intervals and levels. Briefly, a confidence interval is a way of estimating a population parameter that provides an interval of the parameter rather than a single value. A confidence interval is always qualified by a confidence level, usually expressed as a percentage such as 95%. It is an indicator of the reliability of the estimate.

**Z Table from Mean (0 to Z)**

z | 0 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |

0 | 0 | 0.00399 | 0.00798 | 0.01197 | 0.01595 | 0.01994 | 0.02392 | 0.0279 | 0.03188 | 0.03586 |

0.1 | 0.03983 | 0.0438 | 0.04776 | 0.05172 | 0.05567 | 0.05962 | 0.06356 | 0.06749 | 0.07142 | 0.07535 |

0.2 | 0.07926 | 0.08317 | 0.08706 | 0.09095 | 0.09483 | 0.09871 | 0.10257 | 0.10642 | 0.11026 | 0.11409 |

0.3 | 0.11791 | 0.12172 | 0.12552 | 0.1293 | 0.13307 | 0.13683 | 0.14058 | 0.14431 | 0.14803 | 0.15173 |

0.4 | 0.15542 | 0.1591 | 0.16276 | 0.1664 | 0.17003 | 0.17364 | 0.17724 | 0.18082 | 0.18439 | 0.18793 |

0.5 | 0.19146 | 0.19497 | 0.19847 | 0.20194 | 0.2054 | 0.20884 | 0.21226 | 0.21566 | 0.21904 | 0.2224 |

0.6 | 0.22575 | 0.22907 | 0.23237 | 0.23565 | 0.23891 | 0.24215 | 0.24537 | 0.24857 | 0.25175 | 0.2549 |

0.7 | 0.25804 | 0.26115 | 0.26424 | 0.2673 | 0.27035 | 0.27337 | 0.27637 | 0.27935 | 0.2823 | 0.28524 |

0.8 | 0.28814 | 0.29103 | 0.29389 | 0.29673 | 0.29955 | 0.30234 | 0.30511 | 0.30785 | 0.31057 | 0.31327 |

0.9 | 0.31594 | 0.31859 | 0.32121 | 0.32381 | 0.32639 | 0.32894 | 0.33147 | 0.33398 | 0.33646 | 0.33891 |

1 | 0.34134 | 0.34375 | 0.34614 | 0.34849 | 0.35083 | 0.35314 | 0.35543 | 0.35769 | 0.35993 | 0.36214 |

1.1 | 0.36433 | 0.3665 | 0.36864 | 0.37076 | 0.37286 | 0.37493 | 0.37698 | 0.379 | 0.381 | 0.38298 |

1.2 | 0.38493 | 0.38686 | 0.38877 | 0.39065 | 0.39251 | 0.39435 | 0.39617 | 0.39796 | 0.39973 | 0.40147 |

1.3 | 0.4032 | 0.4049 | 0.40658 | 0.40824 | 0.40988 | 0.41149 | 0.41308 | 0.41466 | 0.41621 | 0.41774 |

1.4 | 0.41924 | 0.42073 | 0.4222 | 0.42364 | 0.42507 | 0.42647 | 0.42785 | 0.42922 | 0.43056 | 0.43189 |

1.5 | 0.43319 | 0.43448 | 0.43574 | 0.43699 | 0.43822 | 0.43943 | 0.44062 | 0.44179 | 0.44295 | 0.44408 |

1.6 | 0.4452 | 0.4463 | 0.44738 | 0.44845 | 0.4495 | 0.45053 | 0.45154 | 0.45254 | 0.45352 | 0.45449 |

1.7 | 0.45543 | 0.45637 | 0.45728 | 0.45818 | 0.45907 | 0.45994 | 0.4608 | 0.46164 | 0.46246 | 0.46327 |

1.8 | 0.46407 | 0.46485 | 0.46562 | 0.46638 | 0.46712 | 0.46784 | 0.46856 | 0.46926 | 0.46995 | 0.47062 |

1.9 | 0.47128 | 0.47193 | 0.47257 | 0.4732 | 0.47381 | 0.47441 | 0.475 | 0.47558 | 0.47615 | 0.4767 |

2 | 0.47725 | 0.47778 | 0.47831 | 0.47882 | 0.47932 | 0.47982 | 0.4803 | 0.48077 | 0.48124 | 0.48169 |

2.1 | 0.48214 | 0.48257 | 0.483 | 0.48341 | 0.48382 | 0.48422 | 0.48461 | 0.485 | 0.48537 | 0.48574 |

2.2 | 0.4861 | 0.48645 | 0.48679 | 0.48713 | 0.48745 | 0.48778 | 0.48809 | 0.4884 | 0.4887 | 0.48899 |

2.3 | 0.48928 | 0.48956 | 0.48983 | 0.4901 | 0.49036 | 0.49061 | 0.49086 | 0.49111 | 0.49134 | 0.49158 |

2.4 | 0.4918 | 0.49202 | 0.49224 | 0.49245 | 0.49266 | 0.49286 | 0.49305 | 0.49324 | 0.49343 | 0.49361 |

2.5 | 0.49379 | 0.49396 | 0.49413 | 0.4943 | 0.49446 | 0.49461 | 0.49477 | 0.49492 | 0.49506 | 0.4952 |

2.6 | 0.49534 | 0.49547 | 0.4956 | 0.49573 | 0.49585 | 0.49598 | 0.49609 | 0.49621 | 0.49632 | 0.49643 |

2.7 | 0.49653 | 0.49664 | 0.49674 | 0.49683 | 0.49693 | 0.49702 | 0.49711 | 0.4972 | 0.49728 | 0.49736 |

2.8 | 0.49744 | 0.49752 | 0.4976 | 0.49767 | 0.49774 | 0.49781 | 0.49788 | 0.49795 | 0.49801 | 0.49807 |

2.9 | 0.49813 | 0.49819 | 0.49825 | 0.49831 | 0.49836 | 0.49841 | 0.49846 | 0.49851 | 0.49856 | 0.49861 |

3 | 0.49865 | 0.49869 | 0.49874 | 0.49878 | 0.49882 | 0.49886 | 0.49889 | 0.49893 | 0.49896 | 0.499 |

3.1 | 0.49903 | 0.49906 | 0.4991 | 0.49913 | 0.49916 | 0.49918 | 0.49921 | 0.49924 | 0.49926 | 0.49929 |

3.2 | 0.49931 | 0.49934 | 0.49936 | 0.49938 | 0.4994 | 0.49942 | 0.49944 | 0.49946 | 0.49948 | 0.4995 |

3.3 | 0.49952 | 0.49953 | 0.49955 | 0.49957 | 0.49958 | 0.4996 | 0.49961 | 0.49962 | 0.49964 | 0.49965 |

3.4 | 0.49966 | 0.49968 | 0.49969 | 0.4997 | 0.49971 | 0.49972 | 0.49973 | 0.49974 | 0.49975 | 0.49976 |

3.5 | 0.49977 | 0.49978 | 0.49978 | 0.49979 | 0.4998 | 0.49981 | 0.49981 | 0.49982 | 0.49983 | 0.49983 |

3.6 | 0.49984 | 0.49985 | 0.49985 | 0.49986 | 0.49986 | 0.49987 | 0.49987 | 0.49988 | 0.49988 | 0.49989 |

3.7 | 0.49989 | 0.4999 | 0.4999 | 0.4999 | 0.49991 | 0.49991 | 0.49992 | 0.49992 | 0.49992 | 0.49992 |

3.8 | 0.49993 | 0.49993 | 0.49993 | 0.49994 | 0.49994 | 0.49994 | 0.49994 | 0.49995 | 0.49995 | 0.49995 |

3.9 | 0.49995 | 0.49995 | 0.49996 | 0.49996 | 0.49996 | 0.49996 | 0.49996 | 0.49996 | 0.49997 | 0.49997 |

4 | 0.49997 | 0.49997 | 0.49997 | 0.49997 | 0.49997 | 0.49997 | 0.49998 | 0.49998 | 0.49998 | 0.49998 |

### Related Videos

## Venn Diagrams: Set Notation

### Purplemath

Venn diagrams can be used to express the logical (in the mathematical sense) relationships between various sets. The following examples should help you understand the notation, terminology, and concepts relating Venn diagrams and set notation.

Let's say that our universe contains the numbers 1, 2, 3, and 4, so U = {1, 2, 3, 4}. Let A be the set containing the numbers 1 and 2; that is, A = {1, 2}.

Note: The curly braces are the customary notation for sets. Do *not* use parentheses or square brackets.

### MathHelp.com

Let B be the set containing the numbers 2 and 3; that is, B = {2, 3}. Then we can find various set relationships with the help of Venn diagrams. In what follows, I've used pinkish shading to mark the solution "regions" in the Venn diagrams.

For A = {1, 2}, B = {2, 3}, U = {1, 2, 3, 4}, find the following using a Venn diagram:

is pronounced as: A union B

means: the new set that contains every element from either of A and B; if a thing is in either one of these sets, it's in the new set

in terms of the elements:

Venn diagram:

my answer:

The Venn diagram above illustrates the set notation and the logic of the answer. Since "union" means "everything in either of the sets", all of each circle is shaded in. (If you're not clear on the logic of the set notation, review set notation before proceeding further.)

The following bank of america customer care email work in the same way.

is pronounced as: "A intersect B"

means: the new set that contains every element that is in *both* of the input sets; only things inside both of the input sets get added to the new set

in terms of the elements:

Venn diagram:

my answer:

#### (sometimes denoted as ~A or )

is pronounced as: "A complement" (or "not A", for aub c other notations)

means: the new set gets everything that is in the universe but is outside of A; it's okay if the element is in B, just so long as it is *not* also in A

in terms of the elements: {1, 2, 3, 4} – {1, 2}

Venn diagram:

my answer:

The tilde ("TILL-duh") is the wiggly "~" character at the beginning of ~A; on your keyboard, the tilde is probably located at or near the left-hand end of the row of numbers. The tilde, in the set-relation context, says that I now want to find the complement (in a sense, the opposite) of whatever is being negated or "thrown out"; in *aub c* case, that's the set 1st financial federal credit union routing number. The kind of complement we see in this exercise, the "not" complement, means "throw out everything you have now (in this case, the set A) and take everything else in the universe instead".

Practically speaking, the "not" complement with the tilde says to reverse the shading, which is how I got the final picture above.

is pronounced as: "A minus B" or "A complement B"

means: the new set gets everything that is in A*except for* anything in its overlap with B; if it's in A and *not* in B, then it goes into the new set; nothing from the overlap in the diagram (being the intersection of the input sets) goes into the new set

in terms of the elements: {1, 2} – {2, 3}

Venn diagram:

my answer: A – B = {1}

is pronounced as: "not (A union B)" (or "the complement of (A union B)")

means: the new set gets everything that is outside A and B; if something is in either of the input sets, it doesn't go into the new set

in terms of the elements: {1, 2, 3, 4} – ({1, 2} {2, 3}) = {1, 2, 3, 4} – {1, 2, 3}

Venn diagram:

my answer: ~

is pronounced as: "not (A intersect B)" (or "the complement of (A intersect B)")

means: the new set contains everything outside of the overlap of A and B; if something is in *both* of the input sets, then it isn't in the new set; everything *outside* of the overlap of the input sets (including everything in the universe but outside of the input sets) goes into the new set

in terms of the elements: {1, 2, 3, 4} – ({1, 2} {2, 3}) = {1, 2, 3, 4} – {2}

Venn diagram:

my answer: ~

There are gazillions of other possibilities for set combinations and relationships, but the above are among the simplest and most common. Some of the examples above showed more than one way of formatting (and pronouncing) the same thing. Different texts use different set notation, so you should not be at all surprised *aub c* your text uses still other symbols than those used above. But while the notation may differ, the concepts will be the same.

By the way, as you probably noticed, your Venn-diagram "circles" don't have to be perfectly round; ellipses will do just fine.

Sometimes you'll be asked to find set intersections, unions, etc, without knowing what the sets actually are. This is okay. The Venn diagrams can still help you figure out the set relations.

#### Given the following Venn diagram, shade in .

My thinking: The intersection of A and C is just the overlap between those two circles, so my answer is:

The exercise only asked for a graph of the result of the set notation. They didn't give me any elements for the universe or any of the sets. The shaded-in picture is the answer they're wanting.

#### Given the following Venn diagram, shade in .

My thinking: As usual when faced with parentheses, I'll work from the inside out.

I'll first find B – C. "B complement C" means I take B and then throw out its overlap with C, which gives me this:

Now I have to union this with A, meaning that I have to add everything from A to what I got at the previous step. My answer is:

Note that unioning with A in the last step above put some of C (that is, some of what I'd cut out when I did "B – C") back into the answer. This is okay. Just because we threw out C at one point, doesn't mean that it all has to stay out forever.

#### Given the following Venn diagram, shade in ~[(B C) – A].

As usual when dealing with nested grouping symbols, I'll work from the inside out.

The union of B and C shades both circles fully:

Now I'll do *aub c* "complement A" part by cutting out the overlap with A:

The "not" complement with the tilde says to reverse the shading, so my final answer is:

URL: https://www.purplemath.com/modules/venndiag2.htm

## Aubameyang crucial to Arsenal: Arteta

Arsenal manager Mikel Arteta has backed skipper Pierre-Emerick Aubameyang to rediscover his goalscoring form with the Gabon striker failing to find the net in his last four games.

Aubameyang, the league's joint top scorer in 2018-19, has struggled so far this season with only four goals in 12 appearances.

The 32-year-old has been paired with fellow striker Alexandre Lacazette up front in recent games, with both players tasked with pressing defenders to win the ball back, and Arteta said the duo must adapt to changes in the team's formation.

"The form of Auba is always going to be related to the goals he scores. He does many other things apart from scoring goals, but obviously we need that contribution because goals are critical for our success," Arteta told reporters on Wednesday.

"That's been shown in the last *aub c* years when we've been able to do that and that has to be his aim. I think with Laca it's a little bit different. I think Laca is more comfortable playing (at No.9).

"The formation that we play allows him more freedom to attack certain spaces more often and the defensive duty changes a lot from one formation or the other. That's something to consider, depending on the players that play alongside him."

Arteta said Granit Xhaka and Sead Kolasinac will not be involved in their next EPL match - against Manchester United at Old Trafford on Thursday - but added he was hopeful Bukayo Saka would be available after the forward picked up an injury against Newcastle United at the weekend.

Arteta also praised United's appointment of Ralf Rangnick as interim coach until the end of the season.

"He has been one of the top coaches in every involvement he has had in the game, not only in coaching," Arteta said.

"He's made some radical changes in some areas and he's a really interesting person in the footballing world, a really interesting appointment for the club."

Rangnick, 63, will not be in charge for Thursday's game as his work visa is yet to be finalised, with Michael Carrick remaining as caretaker. United sit five points behind fifth-placed Arsenal.

### Get the latest news from thewest.com.au in your inbox.

Sign up for our emails

Источник: https://thewest.com.au/sport/soccer/aubameyang-crucial-to-arsenal-arteta-c-4779560### Union of A and B

In probability, the union of events, **P(A U B)**, essentially involves the condition where any or all of the events being considered occur, shown in the Venn diagram below. Note that **P(A U B)** can also be written as **P(A OR B)**. In this case, the "inclusive OR" is being used. This means that while at least one of the conditions within the union must hold true, all conditions can be simultaneously true. There are two cases for the union of events; the events are either mutually exclusive, or the events are not mutually exclusive. In the case where the events are mutually exclusive, aub c calculation of the probability is simpler:

A basic example of mutually exclusive events would be the rolling of a dice, where event **A** is the probability that an even number is rolled, and event **B** is the probability that an aub c number is rolled. It is clear in this case that the events are mutually exclusive since a number cannot be both even and odd, so **P(A U B)** would be **3/6 + 3/6 = 1**, since a standard dice only has odd and even numbers.

The calculator above computes the other case, where the events **A** and **B** are not mutually exclusive. In this case:

P(A U B) = P(A) + P(B) - P(A ∩ B)

Using the example of rolling dice again, find the probability that an even number or a number that is a multiple of 3 is rolled. Here the set is represented by the 6 values of the dice, written as:

S = {1,2,3,4,5,6} | |

Probability of an even number: | P(A) = {2,4,6} = 3/6 |

Probability of a multiple of 3: | P(B) = {3,6} = 2/6 |

Intersection of A and B: | P(A ∩ B) = {6} = 1/6 |

P(A U B) = 3/6 + 2/6 -1/6 = 2/3 |

### Exclusive OR of A and B

Another possible scenario that the calculator above computes is **P(A XOR B)**, shown in the Venn diagram below. The "Exclusive OR" operation is defined as the event that A or B occurs, but not simultaneously. The equation is as follows:

As an example, imagine it is Halloween, and two buckets of candy are set outside the house, one rio grande regional hospital careers Snickers, and the other containing Reese's. Multiple flashing neon signs are placed around the buckets of candy insisting that each trick-or-treater only takes one Snickers *Aub c* Reese's but not both! It is unlikely, however, that every child adheres to the flashing neon signs. Given a probability of Reese's being chosen as **P(A) = 0.65**, or Snickers being chosen with **P(B) = 0.349**, and a **P(unlikely) = 0.001** that a child exercises restraint while considering the detriments of a potential future cavity, calculate the probability that Snickers or Reese's is chosen, but not both:

0.65 + 0.349 - 2 × 0.65 × 0.349 = 0.999 - 0.4537 = 0.5453

Therefore, there is a 54.53% chance that Snickers or Reese's is chosen, but not both.

### Normal Distribution

The normal distribution or Gaussian distribution is a continuous probability distribution that follows the function of:

where **μ** is the mean and **σ ^{2}** is the variance. Note that standard deviation is typically denoted as

**σ**. Also, in the special case where

**μ = 0**and

**σ = 1**, the distribution is referred to as a standard normal distribution. Above, along with the calculator, is a diagram of a typical normal distribution curve.

The normal distribution is often used to describe and approximate any variable that tends to cluster around the mean, for example, the heights of male students in a college, the leaf sizes on a tree, the scores of **aub c** test, etc. Use the "Normal Distribution" calculator above to determine the probability of an event with a normal distribution lying between two given values (i.e. **P** in the diagram above); for example, the probability of the height of a male student is between 5 and 6 feet in a college. Finding **P** as shown in the above diagram involves standardizing the two desired values to a z-score by subtracting the given mean and dividing by the standard deviation, as well as using a Z-table to find probabilities for Z. If, for example, it is desired to find the probability that a student at a university has a height between 60 inches and 72 inches tall given a mean of 68 inches tall with a standard deviation of 4 inches, 60 and first financial bank texas customer service number inches would be standardized as such:

Given **μ** = 68; **σ** = 4

(60 - 68)/4 = -8/4 = -2

(72 - 68)/4 = 4/4 = 1

The graph above illustrates the area of interest in the normal distribution. In order to determine the probability represented by the shaded area of the graph, use the standard normal Z-table provided at the bottom of the page. Note that there are different types of standard normal Z-tables. The table below provides the probability that a statistic is between 0 and Z, where 0 is the mean in the standard normal distribution. There are also Z-tables that provide the probabilities left or right of Z, both of which can be used to calculate the desired probability by subtracting the relevant values.

For this example, to determine the probability of a value between 0 and 2, find 2 in the first column of the table, since this table by definition provides probabilities christmas tree in the park san jose the mean (which is 0 in the standard normal distribution) and the number of choices, in this case, 2. Note that since the value in question is 2.0, the table is read by lining up the 2 row with the 0 column, and reading the value therein. If, instead, the value in question were 2.11, the 2.1 row would be matched with the 0.01 column and the value would be 0.48257. Also, note that even though the actual value of interest is -2 on the graph, the table only provides positive values. Since the normal distribution is symmetrical, only the displacement is important, and a displacement of 0 to -2 or 0 to 2 is the same, and will have the same area under the curve. Thus, the probability of a value falling between 0 and 2 is 0.47725while a value between 0 and 1 has a probability of 0.34134. Since the desired area is between -2 and 1, the probabilities are added to yield 0.81859, or approximately 81.859%. Returning to the example, this means that there is an 81.859% chance in this case that a male student at the given university has a height between 60 and 72 inches.

The calculator also provides a table of confidence intervals for various confidence levels. Refer to the Sample Size Calculator for Proportions for a more detailed explanation of confidence intervals and levels. Briefly, a confidence interval is a way of estimating a population parameter that provides an interval of the parameter rather than a single value. A confidence interval is always qualified by a confidence level, usually expressed as a percentage such as 95%. It is an indicator of the reliability of the estimate.

**Z Table from Mean (0 to Z)**

z | 0 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |

0 | 0 | 0.00399 | 0.00798 | 0.01197 | 0.01595 | 0.01994 | 0.02392 | 0.0279 | 0.03188 | 0.03586 |

0.1 | 0.03983 | 0.0438 | 0.04776 | 0.05172 | 0.05567 | 0.05962 | 0.06356 | 0.06749 | 0.07142 | 0.07535 |

0.2 | 0.07926 | 0.08317 | 0.08706 | 0.09095 | 0.09483 | 0.09871 | 0.10257 | 0.10642 | 0.11026 | 0.11409 |

0.3 | 0.11791 | 0.12172 | 0.12552 | 0.1293 | 0.13307 | 0.13683 | 0.14058 | 0.14431 | 0.14803 | 0.15173 |

0.4 | 0.15542 | 0.1591 | 0.16276 | 0.1664 | 0.17003 | 0.17364 | 0.17724 | 0.18082 | 0.18439 | 0.18793 |

0.5 | 0.19146 | 0.19497 | 0.19847 | 0.20194 | 0.2054 | 0.20884 | 0.21226 | 0.21566 | 0.21904 | 0.2224 |

0.6 | 0.22575 | 0.22907 | 0.23237 | 0.23565 | 0.23891 | 0.24215 | 0.24537 | 0.24857 | 0.25175 | 0.2549 |

0.7 | 0.25804 | 0.26115 | 0.26424 | 0.2673 | 0.27035 | 0.27337 | 0.27637 | 0.27935 | 0.2823 | 0.28524 |

0.8 | 0.28814 | 0.29103 | 0.29389 | 0.29673 | 0.29955 | 0.30234 | 0.30511 | 0.30785 | 0.31057 | 0.31327 |

0.9 | 0.31594 | 0.31859 | 0.32121 | 0.32381 | 0.32639 | 0.32894 | 0.33147 | 0.33398 | 0.33646 | 0.33891 |

1 | 0.34134 | 0.34375 | 0.34614 | 0.34849 | 0.35083 | 0.35314 | 0.35543 | 0.35769 | 0.35993 | 0.36214 |

1.1 | 0.36433 | 0.3665 | 0.36864 | 0.37076 | 0.37286 | 0.37493 | 0.37698 | 0.379 | 0.381 | 0.38298 |

1.2 | 0.38493 | 0.38686 | 0.38877 | 0.39065 | 0.39251 | 0.39435 | 0.39617 | 0.39796 | 0.39973 | 0.40147 |

1.3 | 0.4032 | 0.4049 | 0.40658 | 0.40824 | 0.40988 | 0.41149 | 0.41308 | 0.41466 | 0.41621 | 0.41774 |

1.4 | 0.41924 | 0.42073 | 0.4222 | 0.42364 | 0.42507 | 0.42647 | 0.42785 | 0.42922 | 0.43056 | 0.43189 |

1.5 | 0.43319 | 0.43448 | 0.43574 | 0.43699 | 0.43822 | 0.43943 | 0.44062 | 0.44179 | 0.44295 | 0.44408 |

1.6 | 0.4452 | 0.4463 | 0.44738 | 0.44845 | 0.4495 | 0.45053 | 0.45154 | 0.45254 | 0.45352 | 0.45449 |

1.7 | 0.45543 | 0.45637 | 0.45728 | 0.45818 | 0.45907 | 0.45994 | 0.4608 | 0.46164 | 0.46246 | 0.46327 |

1.8 | 0.46407 | 0.46485 | 0.46562 | 0.46638 | 0.46712 | 0.46784 | 0.46856 | 0.46926 | 0.46995 | 0.47062 |

1.9 | 0.47128 | 0.47193 | 0.47257 | 0.4732 | 0.47381 | 0.47441 | 0.475 | 0.47558 | 0.47615 | 0.4767 |

2 | 0.47725 | 0.47778 | 0.47831 | 0.47882 | 0.47932 | 0.47982 | 0.4803 | 0.48077 | 0.48124 | 0.48169 |

2.1 | 0.48214 | 0.48257 | 0.483 | 0.48341 | 0.48382 | 0.48422 | 0.48461 | 0.485 | 0.48537 | 0.48574 |

2.2 | 0.4861 | 0.48645 | 0.48679 | 0.48713 | 0.48745 | 0.48778 | 0.48809 | 0.4884 | 0.4887 | 0.48899 |

2.3 | 0.48928 | 0.48956 | 0.48983 | 0.4901 | 0.49036 | 0.49061 | 0.49086 | 0.49111 | 0.49134 | 0.49158 |

2.4 | 0.4918 | 0.49202 | 0.49224 | 0.49245 | 0.49266 | 0.49286 | 0.49305 | 0.49324 | 0.49343 | 0.49361 |

2.5 | 0.49379 | 0.49396 | 0.49413 | 0.4943 | 0.49446 | 0.49461 | 0.49477 | 0.49492 | 0.49506 | 0.4952 |

2.6 | 0.49534 | 0.49547 | 0.4956 | 0.49573 | 0.49585 | 0.49598 | 0.49609 | 0.49621 | 0.49632 | 0.49643 |

2.7 | 0.49653 | 0.49664 | 0.49674 | 0.49683 | 0.49693 | 0.49702 | 0.49711 | 0.4972 | 0.49728 | 0.49736 |

2.8 | 0.49744 | 0.49752 | 0.4976 | 0.49767 | 0.49774 | 0.49781 | 0.49788 | 0.49795 | 0.49801 | 0.49807 |

2.9 | 0.49813 | 0.49819 | 0.49825 | 0.49831 | 0.49836 | 0.49841 | 0.49846 | 0.49851 | 0.49856 | 0.49861 |

3 | 0.49865 | 0.49869 | 0.49874 | 0.49878 | 0.49882 | 0.49886 | 0.49889 | 0.49893 | 0.49896 | 0.499 |

3.1 | 0.49903 | 0.49906 | 0.4991 | 0.49913 | 0.49916 | 0.49918 | 0.49921 | 0.49924 | 0.49926 | 0.49929 |

3.2 | 0.49931 | 0.49934 | 0.49936 | 0.49938 | 0.4994 | 0.49942 | 0.49944 | 0.49946 | 0.49948 | 0.4995 |

3.3 | 0.49952 | 0.49953 | 0.49955 | 0.49957 | 0.49958 | 0.4996 | 0.49961 | 0.49962 | 0.49964 | 0.49965 |

3.4 | 0.49966 | 0.49968 | 0.49969 | 0.4997 | 0.49971 | 0.49972 | 0.49973 | 0.49974 | 0.49975 | 0.49976 |

3.5 | 0.49977 | 0.49978 | 0.49978 | 0.49979 | 0.4998 | 0.49981 | 0.49981 | 0.49982 | 0.49983 | 0.49983 |

3.6 | 0.49984 | 0.49985 | 0.49985 | 0.49986 | 0.49986 | 0.49987 | 0.49987 | 0.49988 | 0.49988 | 0.49989 |

3.7 | 0.49989 | 0.4999 | 0.4999 | 0.4999 | 0.49991 | 0.49991 | 0.49992 | 0.49992 | 0.49992 | 0.49992 |

3.8 | 0.49993 | 0.49993 | 0.49993 | 0.49994 | 0.49994 | 0.49994 | 0.49994 | 0.49995 | 0.49995 | 0.49995 |

3.9 | 0.49995 | 0.49995 | 0.49996 | 0.49996 | 0.49996 | 0.49996 | 0.49996 | 0.49996 | 0.49997 | 0.49997 |

4 | 0.49997 | 0.49997 | 0.49997 | 0.49997 | 0.49997 | 0.49997 | 0.49998 | 0.49998 | 0.49998 | 0.49998 |