Consider a binomial experiment with 15 trials and probability 0.45 of success on

Consider a binomial experiment with 15 trials and probability 0.45 of success on a single trial.
(a) Use the binomial distribution to find the probability of exactly 10 successes. (Round your answer to three decimal places.)
(b) Use the normal distribution to approximate the probability of exactly 10 successes. (Round your answer to three decimal places.)
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Step 1
Solution: It is given here that a random variable say x follows the binomial distribution with parameters
The binomial probability function is:
$P\left(X=x\right)=\frac{n!}{\left(n-x\right)!x!}{p}^{x}{\left(1-p\right)}^{n-x};x=0,1,2,..,n$
Step 2
(a) Use the binomial distribution to find the probability of exactly 10 successes.
Answer: It is required to find:
$P\left(x=10\right)$
Using the binomial distribution function:
$P\left(x=10\right)=\frac{15!}{\left(15-10\right)!10!}{0.45}^{10}{\left(1-0.45\right)}^{15-10}$
$=3003×0.000340506×0.050328438$
$=0.051$
Therefore, the probability of exactly 10 successes is 0.051
Step 3
(b) Use the normal distribution to approximate the probability of exactly 10 successes.
The mean and standard deviation of the random variable x is:
$\mu =np=15×0.45=6.75$
$\sigma =\sqrt{np\left(1-p\right)}=\sqrt{15×0.45\left(1-0.45\right)}=1.92678$
It is required to find:
$P\left(x=10\right)$
Using the continuity correction factor, the above probability can be written as:
$P\left(x=10\right)=P\left(10-0.5
$=P\left(9.5
Using the z-score formula:
$P\left(9.5
$=P\left(1.4272
$=P\left(z<1.9462\right)-P\left(z<1.4272\right)$
Now using the excel functions:
$P\left(9.5
The excel functions are:
$=NORMSDIST\left(1.4272\right)=0.9232$
$=NORMSDIST\left(1.9462\right)=0.9742$
Therefore, Using the normal distribution to approximate the probability of exactly 10 successes is 0.051