Question

136 cars were sold during the month of April. 71 had air conditioning and 72 had automatic transmission. 55 had air conditioning only, 56 had automatic transmission only, and 9 had neither of these extras. What is the probability that a randomly selected car had automatic transmission or air conditioning or both

Answer 1

0.93382

Explanation:

Let

The event where a car has air conditioning (ac) be A

The event where a car has an automatic transmission be B

`P(\text{Car has ac and cars that don't have automatic transmission})=P(A\cap B')=(71)/(136)`

`P(\text{Car has car has an automatic transmission and no ac})=P(B\cap A')=(56)/(136)`

`P(\text{Car has no ac and cars that don't have automatic transmission})=P(A'\cap B')=(9)/(136)`

The probability of a chosen car which has ac or automatic transmission or both

`P(A\cup B)=1-P(A\cup B)'\\\Rightarrow P(A\cup B)=1-P(A'\cap B')\\\Rightarrow P(A\cup B)=1-(9)/(136)\\\Rightarrow P(A\cup B)=0.93382`

The probability that a randomly selected car had automatic transmission or air conditioning or both is 0.93382