91Ó°ÊÓ

Let X be the total number of successes in 15 Bernoulli trails.

The probability of success is p=0.3

The probability of success is p=0.3 in 15 trials

Hence

\(\begin{array}{c}E\left( X \right) = 15\left( {0.3} \right)\\ = 4.5\end{array}\)

And variance is

\(\begin{array}{c}\sigma = E{\left( X \right)^2} - \left[ {E\left( X \right)} \right]\\ = {\left[ {\left( {15} \right)\left( {0.3} \right)\left( {0.7} \right)} \right]^{\frac{1}{2}}}\\ = 1.775\end{array}\)

Therefore,

\(\begin{array}{c}{\rm P}\left( {X = 4} \right) = {\rm P}\left( {3.5 \le X \le 4.5} \right)\\ = {\rm P}\left( {\frac{{3.5 - 4.5}}{{1.775}} \le Z \le 0} \right)\end{array}\)

\(\begin{array}{c}{\rm P}\left( {X = 4} \right) = {\rm P}\left( {\frac{{ - 1}}{{1.775}} \le Z \le 0} \right)\\ = {\rm P}\left( { - 5.634 \le Z \le 0} \right)\end{array}\)

\(\begin{array}{c}{\rm P}\left( {X = 4} \right) \approx \phi \left( {0.5634} \right) - 0.5\\ \approx 0.214\end{array}\)

Hence Approximate value of \({\rm P}\left( {X = 4} \right)\) is 0.214

The exact value from the table of binomial probabilities for\(\left( {n = 15,p = 0.3,k = 4} \right)\)is found to be 0.2186

Hence the approximate value of\({\rm P}\left( {X = 4} \right)\)is 0.214 and

The exact value of \({\rm P}\left( {X = 4} \right)\) is 0.2186