Complete the table and construct the probability histogram for a

Kelly Nelson

Kelly Nelson

Answered question

2021-12-12

Complete the table and construct the probability histogram for a binomial random variable x with n=6 and p=0.5.
x0123456p(x)

Answer & Explanation

Karen Robbins

Karen Robbins

Beginner2021-12-13Added 49 answers

We know that,
P.d.f of binomial distribution is given by,
p(x)=nCxpxqnx
Where q=1p
If Xbomial(6,0.5)
Then p(x)=6Cx(0.5)x(0.5)6x
For x=0
p(x=0)=6C0(0.5)0(0.5)60
=0.015625
Similarly,
p(x=1)=0.09375
p(x=2)=0.234375
p(x=3)=0.3125
p(x=4)=0.234375
p(x=5)=0.09375
p(x=6)=0.015625
Jonathan Burroughs

Jonathan Burroughs

Beginner2021-12-14Added 37 answers

Step 1
P(1) Probability of exactly 1 successes
If using a calculator, you can enter trials=6
p=0.5
andX=1
into a binomial probability distribution function (PDF). If doing this by hand, apply the binomial probability formula:
P(X)=(nx)×px×(1p)nx
The binimial 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 of success on a single trial, and X is the number of successes. Substituting in values for this problem, n=6, p=0.5, and X=1.
P(1)=6!1!(61)!×0.51×(10.5)61
Evaluating the expression, we have
P(1)=0.09375
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=0.5 and 6 trials is:
P(0)=0.015625
P(1)=0.09375
P(2)=0.234375
P(3)=0.3125
P(4)=0.234375
P(5)=0.09375
P(6)=0.015625

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