In a certain city, incompatibility is given as the legal reason in 70 percent of all divorce cases. Find the probability that five of the next six divorce cases filed in this city will claim incompatibility as the reason, using

(a) the formula for the binomial distribution;
(b) Table I of "Statistical Tables."

Solution

Let $X$ number of divorce cases who gave incompatibility as a legal reason out of six cases. In a certain city, incompatibility is given as the legal reason in 70 percent of all divorce cases. That is $p = 0.70$.

Here $n = 6$ and $p=0.7$. The probability distribution of $X$ is Binomial distribution. That is $X\sim B(6,0.7)$.

The probability mass function of $X$ is

$$ \begin{aligned} P(X=x) &= \binom{6}{x} (0.7)^x (1-0.7)^{6-x},\\ & \qquad x=0,1,\cdots, 6. \end{aligned} $$

(a) The probability that $X$ is exactly $5$ is

$$ \begin{aligned} P(X= 5) & =\binom{6}{5} (0.7)^{5} (1-0.7)^{6-5}\\ & = 0.3025\\ \end{aligned} $$

The probability that five of the next six divorce cases filed in this city will claim incompatibility as the reason is $P(X=5) = 0.3025$.

(b) Using Binomial Statistical table

$$ \begin{aligned} P(X= 5) &= P(X\leq 5) - P(X\leq 4)\\ &= 0.8824 - 0.5798\\ &= 0.3026 \end{aligned} $$

The probability that five of the next six divorce cases filed in this city will claim incompatibility as the reason is $P(X=5) = 0.3026$.