You gather a random sample of six inmates who are eligible for a parole hearing. The parole hearing has two possible outcomes (i.e., released on parole and not released on parole). You want to construct a binomial probability distribution to be able to later on (after you learn how many of the six inmates were actually released on parole) find whether the actual outcome could have been expected.
a. Create a binomial probability distribution.
b. What is the (theoretical) probability that four of six inmates get released on parole?

Question

Quizplus · Accepted Answer

The Answer is a.

\begin{array} { | l | l | c | l | } \hline { r } & n C r & \left( p ^ { n } ight) \left( q ^ { n - r } ight) & { p } \\hline 0 ext { parole } & 1 & ( 1 / 2 ) ^ { 0 } ( 1 / 2 ) ^ { 6 } = 1 / 64 & 1 / 64 = 0.015625 \\hline 1 & 6 & ( 1 / 2 ) ^ { 1 } ( 1 / 2 ) ^ { 5 } = 1 / 64 & 6 / 64 = 0.9375 \\hline 2 & 15 & ( 1 / 2 ) ^ { 2 } ( 1 / 2 ) ^ { 4 } = 1 / 64 & 15 / 64 = 0.234375 \\hline 3 & 20 & ( 1 / 2 ) ^ { 3 } ( 1 / 2 ) ^ { 3 } = 1 / 64 & 20 / 64 = 0.3125 \\hline 4 & 15 & ( 1 / 2 ) ^ { 4 } ( 1 / 2 ) ^ { 2 } = 1 / 64 & 15 / 64 = 0.234375 \\hline 5 & 6 & ( 1 / 2 ) ^ { 5 } ( 1 / 2 ) ^ { 1 } = 1 / 64 & 6 / 64 = 0.9375 \\hline 6 & 1 & ( 1 / 2 ) ^ { 6 } ( 1 / 2 ) ^ { 0 } = 1 / 64 & 1 / 64 = 0.015625 \\hline\end{array}

b. The chance that four of six inmates get released on parole is 0.2343, or 23.43%.

You Gather a Random Sample of Six Inmates Who Are