Question

At a certain school, twenty-five percent of the students wear a watch and thirty percent wear a bracelet. Sixty percent of the students wear neither a watch nor a bracelet. (a) One of the students is chosen at random. What is the probability that this student is wearing a watch or a bracelet? (b) What is the probability that this student is wearing both a watch and a bracelet?

Answer 1

Explanation:

Given that at a certain school, twenty-five percent of the students wear a watch and thirty percent wear a bracelet.

A- people who wear watch = 25%

B - people who wear bracelet = 30%

(AUB)' - People who wear neither a watch nor a bracelet=60%

`A \bigcap B` - People who wear both =100%-60% = 40%

a)
`P(AUB) = P(A)+P(B)-P(AB) = 25%+30%-40%\\= 15%`

b) the probability that this student is wearing both a watch and a bracelet

=
`P(A \bigcap B) = 40%`

Answer 2

Answer: a) 0.40 b) 0.15

Explanation:

Let A denotes the event that students wear a watch and B denotes the event that students wear a bracelet.

Given : P(A)=0.25 ; P(B)=0.30

`P(A'\cup B')=0.60`

Since,
`P(A\cup B)=1-P(A'\cup B')=1-0.60=0.40`

Thus, the probability that this student is wearing a watch or a bracelet = 0.40

Also,
`P(A\cup B)=P(A)+P(B)-P(A\cap B)`

`P(A\cap B)=P(A)+P(B)-P(A\cup B)`

`P(A\cap B)=0.25+0.30-0.40\\\\\Rightarrow\ P(A\cap B)=0.15`

Thus, the probability that this student is wearing both a watch and a bracelet= 0.15