Question

At Tech Express, 40% of the customers purchases small coffee and 60% purchases medium. Of those customers purchasing small coffee, 30% prefer decaf. Of those purchasing medium coffee, 50% prefer decaf.
(a) What is the probability that the next customer will request medium and decaf coffee?
(b) What is the probability that the next customer prefers decaf?
(c) If the next customer prefers decaf, what is the probability that small is requested?

Answer 1

Answer and explanation:

Given : At Tech Express, 40% of the customers purchases small coffee and 60% purchases medium. Of those customers purchasing small coffee, 30% prefer decaf. Of those purchasing medium coffee, 50% prefer decaf.

Let $A_1$ customer purchase small coffee.

i.e. $P(A_1)=40\%=0.4$

$A_2$ customer purchase medium coffee.

i.e. $P(A_2)=60\%=0.6$

Let B be the customer purchase prefer decaf.

So, $P(B|A_1)=30\%=0.3$

$P(B|A_2)=50\%=0.5$

(a) What is the probability that the next customer will request medium and decaf coffee?

i.e. $P(A_2\cap B)=P(A_2)\times P(B|A_2)$

$P(A_2\cap B)=0.6\times 0.5$

$P(A_2\cap B)=0.3$

(b) What is the probability that the next customer prefers decaf?

i.e. $P(B)=P(A_1\cap B)+P(A_2\cap B)$

$P(B)=P(A_1)\times P(B|A_1)+P(A_2)\times P(B|A_2)$

$P(B)=0.4\times 0.3+0.6\times 0.5$

$P(B)=0.12+0.3$

$P(B)=0.42$

(c) If the next customer prefers decaf, what is the probability that small is requested?

i.e. $P(A_1|B)=\frac{P(A_1\cap B)}{P(B)}$

$P(A_1|B)=\frac{0.12}{0.42}$

$P(A_1|B)=0.28$