Question

at a schools open house, t - shirts and sweatshirts were sold. each item was purchased by either a student or a parent. the two - way frequency table summarizes a random sample of 70 items sold that night.

	student	parent
sweatshirt	17	14

let t - shirt be the event that an item randomly chosen from among those sold was a t - shirt. let parent be the event that an item randomly chosen from among those sold was purchased by a parent. find the following probabilities. write your answers as decimals.
(a) p(parent) =
(b) p(t - shirt and parent) =
(c) p(t - shirt | parent) =

Explanation:

Step1: Calculate P(parent)

The number of items purchased by parents is \(21 + 14=35\). The total number of items sold is \(70\). So \(P(\text{parent})=\frac{35}{70}=0.5\).

Step2: Calculate P(T - shirt and parent)

The number of T - shirts purchased by parents is \(21\). The total number of items sold is \(70\). So \(P(\text{T - shirt and parent})=\frac{21}{70}=0.3\).

Step3: Calculate P(T - shirt | parent)

By the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here \(A\) is the event of a T - shirt and \(B\) is the event of a parent. We know \(P(\text{T - shirt and parent}) = 0.3\) and \(P(\text{parent})=0.5\). So \(P(\text{T - shirt}|\text{parent})=\frac{0.3}{0.5}=0.6\).

Answer:

(a) \(0.5\)
(b) \(0.3\)
(c) \(0.6\)