Question

the windshield wipers on a car have not been working properly. the probability that the car needs a new motor is 0.55, the probability that the car needs a new switch is 0.4, and the probability that the car needs both is 0.2. what is the probability that the car needs neither a new motor nor a new switch? the probability that the car requires needs neither a new motor nor a new switch is (type an integer or a decimal.)

Explanation:

Step1: Use the formula for \(P(A\cup B)\)

The formula for \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Let \(A\) be the event that the car needs a new motor (\(P(A) = 0.55\)), \(B\) be the event that the car needs a new switch (\(P(B)=0.4\)), and \(P(A\cap B) = 0.2\). Then \(P(A\cup B)=0.55 + 0.4-0.2\).

$$P(A\cup B)=0.55+0.4 - 0.2=0.75$$

Step2: Use the complement rule

The probability that the car needs neither a new motor nor a new switch is \(P(\overline{A\cup B})\). By the complement rule \(P(\overline{A\cup B})=1 - P(A\cup B)\). Since \(P(A\cup B) = 0.75\), then \(P(\overline{A\cup B})=1-0.75\).

$$P(\overline{A\cup B})=1 - 0.75=0.25$$

Answer:

\(0.25\)