91Ó°ÊÓ

The random variable X follows the binomial distribution with parameters \(n = 15\) and \(p = 0.5\)

The probability function of the binomially distributed random variable X is given as,

\(f\left( x \right) = \left\{ \begin{array}{l}\left( \begin{array}{l}n\\x\end{array} \right){p^x}{\left( {1 - p} \right)^{n - x}};\;x = 0,1,...,n\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;;\;otherwise\end{array} \right.\)

where there are n trials with p probability of success in each trial.

The required probability is computed as,

\(\begin{aligned}{c}P\left( {X < 6} \right) = P\left( {X = 0} \right) + P\left( {X = 1} \right) + P\left( {X = 2} \right) + ... + P\left( {X = 5} \right)\\ &= \left( {\left( \begin{aligned}{l}15\\0\end{aligned} \right){{\left( {0.5} \right)}^0}{{\left( {1 - 0.5} \right)}^{15 - 0}}} \right) + \left( {\left( \begin{aligned}{l}15\\1\end{aligned} \right){{\left( {0.5} \right)}^1}{{\left( {1 - 0.5} \right)}^{15 - 1}}} \right) + ... + \left( {\left( \begin{aligned}{l}15\\5\end{aligned} \right){{\left( {0.5} \right)}^5}{{\left( {1 - 0.5} \right)}^{15 - 5}}} \right)\\ &= 0.00003 + 0.00046 + ... + 0.09164\\& = 0.1509\end{aligned}\)

Therefore, the required probability is approximately 0.1509.