Suppose That a Family Has 3 Children Also Suppose That the Probabolity

Probability Figurer

Probability of Ii Events

To observe out the matrimony, intersection, and other related probabilities of two independent events.

Probability of A: P(A)
Probability of B: P(B)

Please input values between 0 and 1.

Probability Solver for Two Events

Please provide whatever ii values below to summate the rest probabilities of two contained events.

Probability of A: P(A)
Probability of B: P(B)
Probability of A Not occuring: P(A')
Probability of B NOT occuring: P(B')
Probability of A and B both occuring: P(A∩B)
Probability that A or B or both occur: P(A∪B)
Probability that A or B occurs only Non both: P(AΔB)
Probability of neither A nor B occuring: P((A∪B)')

Delight input values between 0 and i.

Probability of a Series of Contained Events

	Probability	Echo Times
Result A
Event B

Probability of a Normal Distribution

Use the calculator below to detect the area P shown in the normal distribution, also every bit the confidence intervals for a range of confidence levels.

Mean: (µ)
Standard Deviation (σ):
Left Bound (Lb):		For negative infinite, use -inf
Right Spring (Rb):		For positive infinite, use inf

Probability of Two Events

Probability is the measure of the likelihood of an event occurring. It is quantified every bit a number betwixt 0 and 1, with ane signifying certainty, and 0 signifying that the result cannot occur. It follows that the higher the probability of an issue, the more sure information technology is that the consequence volition occur. In its most general case, probability can be defined numerically as the number of desired outcomes divided by the full number of outcomes. This is farther affected by whether the events being studied are independent, mutually exclusive, or conditional, among other things. The calculator provided computes the probability that an event A or B does not occur, the probability A and/or B occur when they are not mutually exclusive, the probability that both outcome A and B occur, and the probability that either outcome A or event B occurs, but not both.

Complement of A and B

Given a probability A, denoted by P(A), it is simple to summate the complement, or the probability that the result described by P(A) does non occur, P(A'). If, for example, P(A) = 0.65 represents the probability that Bob does non do his homework, his teacher Sally tin predict the probability that Bob does his homework every bit follows:

P(A') = 1 - P(A) = 1 - 0.65 = 0.35

Given this scenario, there is, therefore, a 35% chance that Bob does his homework. Any P(B') would exist calculated in the same way, and it is worth noting that in the reckoner above, can be independent; i.e. if P(A) = 0.65, P(B) does not necessarily have to equal 0.35, and can equal 0.30 or another number.

Intersection of A and B

The intersection of events A and B, written as P(A ∩ B) or P(A AND B) is the joint probability of at least ii events, shown beneath in a Venn diagram. In the instance where A and B are mutually exclusive events, P(A ∩ B) = 0. Consider the probability of rolling a four and 6 on a unmarried coil of a dice; it is not possible. These events would therefore be considered mutually exclusive. Computing P(A ∩ B) is simple if the events are contained. In this instance, the probabilities of events A and B are multiplied. To find the probability that ii separate rolls of a die result in half dozen each fourth dimension:

The calculator provided considers the instance where the probabilities are independent. Calculating the probability is slightly more involved when the events are dependent, and involves an understanding of conditional probability, or the probability of issue A given that result B has occurred, P(A|B). Accept the example of a pocketbook of 10 marbles, 7 of which are blackness, and three of which are blue. Summate the probability of drawing a black marble if a blueish 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 blackness marble:

P(B) = 7/10

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

P(B|A) = 7/9

As can be seen, the probability that a black marble is drawn is affected by any previous upshot where a black or blueish 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 in a higher place:

P(A ∩ B) = P(A) × 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 status where any or all of the events being considered occur, shown in the Venn diagram beneath. 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 ane of the conditions within the marriage must hold true, all atmospheric condition can be simultaneously truthful. In that location are 2 cases for the matrimony of events; the events are either mutually sectional, or the events are non mutually exclusive. In the case where the events are mutually exclusive, the calculation of the probability is simpler:

A basic instance of mutually exclusive events would exist 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 example that the events are mutually exclusive since a number cannot be both even and odd, and so P(A U B) would be iii/6 + 3/6 = 1, since a standard die only has odd and fifty-fifty numbers.

The calculator to a higher place computes the other case, where the events A and B are not mutually exclusive. In this example:

P(A U B) = P(A) + P(B) - P(A ∩ B)

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

	S = {one,2,three,iv,v,half dozen}
Probability of an even number:	P(A) = {ii,4,6} = three/6
Probability of a multiple of three:	P(B) = {3,6} = ii/6
Intersection of A and B:	P(A ∩ B) = {vi} = 1/6
	P(A U B) = 3/6 + 2/6 -1/half dozen = two/iii

Exclusive OR of A and B

Another possible scenario that the calculator higher up computes is P(A XOR B), shown in the Venn diagram below. The "Exclusive OR" functioning is defined every bit the event that A or B occurs, but non simultaneously. The equation is every bit follows:

Equally an instance, imagine it is Halloween, and two buckets of candy are gear up outside the house, one containing Snickers, and the other containing Reese's. Multiple flashing neon signs are placed around the buckets of processed insisting that each play a joke on-or-treater only takes 1 Snickers OR Reese'southward merely not both! It is unlikely, however, that every child adheres to the flashing neon signs. Given a probability of Reese's beingness chosen as P(A) = 0.65, or Snickers existence 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, summate the probability that Snickers or Reese's is chosen, just non both:

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

Therefore, in that location is a 54.53% chance that Snickers or Reese's is called, 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 hateful and σⁱⁱ is the variance. Note that standard deviation is typically denoted as σ. Also, in the special case where μ = 0 and σ = ane, the distribution is referred to as a standard normal distribution. Higher up, along with the calculator, is a diagram of a typical normal distribution curve.

The normal distribution is oftentimes used to draw and estimate any variable that tends to cluster around the mean, for instance, the heights of male students in a college, the leaf sizes on a tree, the scores of a exam, etc. Use the "Normal Distribution" computer above to determine the probability of an event with a normal distribution lying between two given values (i.east. P in the diagram in a higher place); for example, the probability of the height of a male student is betwixt 5 and 6 anxiety in a college. Finding P as shown in the higher up diagram involves standardizing the ii desired values to a z-score past subtracting the given hateful and dividing by the standard deviation, besides as using a Z-tabular array to find probabilities for Z. If, for example, it is desired to find the probability that a student at a university has a peak between threescore inches and 72 inches alpine given a hateful of 68 inches tall with a standard deviation of 4 inches, 60 and 72 inches would be standardized every bit such:

Given μ = 68; σ = 4
(sixty - 68)/4 = -8/4 = -2
(72 - 68)/4 = four/4 = ane

The graph higher up illustrates the area of interest in the normal distribution. In order to determine the probability represented by the shaded expanse of the graph, use the standard normal Z-table provided at the lesser of the folio. 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 hateful in the standard normal distribution. There are likewise Z-tables that provide the probabilities left or right of Z, both of which tin can be used to summate the desired probability by subtracting the relevant values.

For this instance, to determine the probability of a value between 0 and 2, find 2 in the first column of the tabular array, since this tabular array past definition provides probabilities between the hateful (which is 0 in the standard normal distribution) and the number of choices, in this case, 2. Annotation that since the value in question is 2.0, the tabular array is read by lining upward the 2 row with the 0 column, and reading the value therein. If, instead, the value in question were 2.xi, 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 tabular array just provides positive values. Since the normal distribution is symmetrical, only the displacement is of import, and a displacement of 0 to -2 or 0 to 2 is the aforementioned, and will have the same area nether the curve. Thus, the probability of a value falling between 0 and 2 is 0.47725 , while a value betwixt 0 and 1 has a probability of 0.34134. Since the desired area is between -two and ane, the probabilities are added to yield 0.81859, or approximately 81.859%. Returning to the example, this means that at that place is an 81.859% chance in this case that a male pupil at the given university has a peak between 60 and 72 inches.

The calculator also provides a table of confidence intervals for diverse conviction levels. Refer to the Sample Size Calculator for Proportions for a more than detailed explanation of confidence intervals and levels. Briefly, a confidence interval is a mode of estimating a population parameter that provides an interval of the parameter rather than a unmarried 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 guess.

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.ane	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.vii	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.ix	0.31594	0.31859	0.32121	0.32381	0.32639	0.32894	0.33147	0.33398	0.33646	0.33891
one	0.34134	0.34375	0.34614	0.34849	0.35083	0.35314	0.35543	0.35769	0.35993	0.36214
one.i	0.36433	0.3665	0.36864	0.37076	0.37286	0.37493	0.37698	0.379	0.381	0.38298
1.two	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
ane.4	0.41924	0.42073	0.4222	0.42364	0.42507	0.42647	0.42785	0.42922	0.43056	0.43189
one.5	0.43319	0.43448	0.43574	0.43699	0.43822	0.43943	0.44062	0.44179	0.44295	0.44408
1.half-dozen	0.4452	0.4463	0.44738	0.44845	0.4495	0.45053	0.45154	0.45254	0.45352	0.45449
ane.vii	0.45543	0.45637	0.45728	0.45818	0.45907	0.45994	0.4608	0.46164	0.46246	0.46327
i.eight	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
two.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
ii.three	0.48928	0.48956	0.48983	0.4901	0.49036	0.49061	0.49086	0.49111	0.49134	0.49158
ii.4	0.4918	0.49202	0.49224	0.49245	0.49266	0.49286	0.49305	0.49324	0.49343	0.49361
ii.5	0.49379	0.49396	0.49413	0.4943	0.49446	0.49461	0.49477	0.49492	0.49506	0.4952
2.vi	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
ii.8	0.49744	0.49752	0.4976	0.49767	0.49774	0.49781	0.49788	0.49795	0.49801	0.49807
two.ix	0.49813	0.49819	0.49825	0.49831	0.49836	0.49841	0.49846	0.49851	0.49856	0.49861
iii	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
iii.half-dozen	0.49984	0.49985	0.49985	0.49986	0.49986	0.49987	0.49987	0.49988	0.49988	0.49989
iii.7	0.49989	0.4999	0.4999	0.4999	0.49991	0.49991	0.49992	0.49992	0.49992	0.49992
iii.viii	0.49993	0.49993	0.49993	0.49994	0.49994	0.49994	0.49994	0.49995	0.49995	0.49995
iii.9	0.49995	0.49995	0.49996	0.49996	0.49996	0.49996	0.49996	0.49996	0.49997	0.49997
four	0.49997	0.49997	0.49997	0.49997	0.49997	0.49997	0.49998	0.49998	0.49998	0.49998

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