An event has a probability p = \frac{5}{8}. Find

Karen Simpson 2021-12-12 Answered
An event has a probability p=58. Find the complete binomial distribution for n=6 trials.
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Expert Answer

Donald Cheek
Answered 2021-12-13 Author has 41 answers
Given
p=58=0.625
n=6
The binomial distribution is given by
P(X=x)=(nx)pxqnx
P(X=x)=(6x)(0.625)x(10.625)6x
Answer
The probability distribution table is given as under
xp(x)00.00278110.02780920.11587130.25749240.32186550.21457760.059605
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alexandrebaud43
Answered 2021-12-14 Author has 36 answers
Step 1
P(0) Probability of exactly 0 successes
If using a calculator, you can enter trials=6, p=0.625, and X=0 into a binomial probability distribution function (PDF). If doing this by hand, apply the binomial probability formula:
P(X)=(nx)×px×(1p)nx
The binomial coefficient, (nx) is defined by
(nx)=n!X!(nX)!
The full binomial probability formula with the binomial coefficient is
P(X)=n!X!(nX)!×px×(1p)nx
Where n is the number of trials, p is the probability if success on a single trial, and X is the number of successes. Substituting in values for this problem, n=6, p=0.625, and X=0
P(0)=6!0!(60)!×0.6250×(10.625)60
Evaluting the expression, we have
P(0)=0.0027809143066406
Step 2
If we apply the binomial probability formula, or a calculator's binomial probability distribution (PDF) function, to all possible values of X for 6 trials, we can construct a complete binomial distribution table. The sum of the probabilities in this table will always be 1. The complete binomial distribution table for this problem, with p=0.625 and 6 trials is:
P(0)=0.0027809143066406
P(1)=0.027809143066406
P(2)=0.11587142944336
P(3)=0.25749206542969
P(4)=0.32186508178711
P(5)=0.21457672119141
P(6)=0.059604644775391
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