An event has a probability $p=\frac{5}{8}$ . Find the complete binomial distribution for $n=6$ trials.

Karen Simpson
2021-12-12
Answered

An event has a probability $p=\frac{5}{8}$ . Find the complete binomial distribution for $n=6$ trials.

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Donald Cheek

Answered 2021-12-13
Author has **41** answers

Given

$p=\frac{5}{8}=0.625$

$n=6$

The binomial distribution is given by

$$P(X=x)=(\begin{array}{c}n\\ x\end{array}){p}^{x}{q}^{n-x}$$

$$P(X=x)=(\begin{array}{c}6\\ x\end{array})(0.625{)}^{x}(1-0.625{)}^{6-x}$$

Answer

The probability distribution table is given as under

$$\begin{array}{|cc|}\hline x& p(x)\\ 0& 0.002781\\ 1& 0.027809\\ 2& 0.115871\\ 3& 0.257492\\ 4& 0.321865\\ 5& 0.214577\\ 6& 0.059605\\ \hline\end{array}$$

The binomial distribution is given by

Answer

The probability distribution table is given as under

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$\text{trials}=6,\text{}p=0.625,$ and $X=0$ into a binomial probability distribution function (PDF). If doing this by hand, apply the binomial probability formula:

$$P(X)=(\begin{array}{c}n\\ x\end{array})\times {p}^{x}\times (1-p{)}^{n-x}$$

The binomial coefficient,$$(\begin{array}{c}n\\ x\end{array})$$ is defined by

$$(\begin{array}{c}n\\ x\end{array})=\frac{n!}{X!(n-X)!}$$

The full binomial probability formula with the binomial coefficient is

$P\left(X\right)=\frac{n!}{X!(n-X)!}\times {p}^{x}\times {(1-p)}^{n-x}$

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,$n=6,\text{}p=0.625,$ and $X=0$

$P\left(0\right)=\frac{6!}{0!(6-0)!}\times {0.625}^{0}\times {(1-0.625)}^{6-0}$

Evaluting the expression, we have

$P\left(0\right)=0.0027809143066406$

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:

$P\left(0\right)=0.0027809143066406$

$P\left(1\right)=0.027809143066406$

$P\left(2\right)=0.11587142944336$

$P\left(3\right)=0.25749206542969$

$P\left(4\right)=0.32186508178711$

$P\left(5\right)=0.21457672119141$

$P\left(6\right)=0.059604644775391$

P(0) Probability of exactly 0 successes

If using a calculator, you can enter

The binomial coefficient,

The full binomial probability formula with the binomial coefficient is

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,

Evaluting the expression, we have

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

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