Question

The number of email responses was twice the number of phone responses. If a person who preferred a picnic is selected at random, what is the probability that the person responded by email? 20% who preferred a picnic responded by phone and 5% responded by email

Answer 1

Answer:

The probability that a randomly selected person who preferred picnic responded by email is 0.3333.

Step-by-step explanation:

Let A = a person responded by phone, B = a person responded by email and X = a person prefers picnic.

Given:

$n(B)=2\times n(A)\\N=n(A)+n(B)\\=n(A)+2n(A)\\=3n(A)$

$P(X|A)= 0.20\\P(X|B)=0.05$

The probability that a response was through email is:

$P(B)=\frac{n(B)}{n(A)}\\=\frac{2n(A)}{3n(A)}\\= \frac{2}{3}$

Then the probability that a response was through phone is:

$P(A)=1-P(A)\\=1-\frac{2}{3}\\ =\frac{1}{3}$

Compute the probability that a person prefers picnic:

$P(X)=P(X|A)P(A) +P(X|B)P(B)\\=(0.20\times\frac{1}{3})+(0.05\times\frac{2}{3})\\=0.10$

Determine the probability that a randomly selected person who preferred picnic responded by email using the conditional probability as follows:

$P(B|X)=\frac{P(X|B)P(B)}{P(X)}\\= \frac{0.05\times\frac{2}{3} }{0.10}\\=0.3333$

Thus, the probability that a person who prefers picnic responded by email is 0.3333.