Guilt in decision making. The Journal of Behavioral

Step 1 of 3

(a)

We are asked to find the \(P(x=5)\)

Here, random variable X follows a hypergeometric distribution, because we are selecting 10 participants from 57 guilty-state participants of which 45 chose the stated option.

A random variable X is said to have a  hypergeometric probability distribution if and only if

\(p(x)=P(X=x)=\frac{(r, x) \times(N-r, n-x)}{(N, n)} \quad\left[(r, x)=C_{x}^{r}=\frac{n !}{x ! \times(n-x) !}\right] \dots \dots (1)\)

Where y is an integer 0, 1, 2,...........n, subject to the restrictions \(x \leq r\) and \(n-x \leq N-r\).

We have given N = 57, n = 10, and r = 45

Hence using equation (1), we can write,

\(\begin{array}{c}
p(5)=P(X=5)=\frac{(r, 5) \times(N-r, n-5)}{(N, n)} \\
p(5)=P(X=5)=\frac{(45,5) \times(57-45,10-5)}{(57,10)} \\
p(5)=P(X=5)=\frac{C_{5}^{45} \times C_{5}^{2}}{C_{10}^{5}}=0.0224
\end{array}\)

Hence the \(P(x=5)=0.0224 \text {. }\)