Question

the table below shows the number of survey subjects who have received and not received a speeding ticket in the last year, and the color of their cars.

find the probability that a randomly chosen person:
a) has a red car.
b) has a speeding ticket.
c) has a speeding ticket given they have a red car.
d) has a red car given they have a speeding ticket.
e) has a red car and got a speeding ticket.
f) has a red car or got a speeding ticket.

write your answers in decimal form, rounded to the nearest thousandth.

Explanation:

Step1: Probability formula

The probability formula is \(P(A)=\frac{n(A)}{n(S)}\), where \(n(A)\) is the number of elements in event \(A\) and \(n(S)\) is the total number of elements in the sample space.

Step2: Part (a)

For the probability of having a red car, \(n(A) = 280\) (total number of red - car owners) and \(n(S)=563\). So \(P(\text{Red Car})=\frac{280}{563}\approx0.497\)

Step3: Part (b)

For the probability of having a speeding ticket, \(n(A) = 258\) (total number of speeding - ticket receivers) and \(n(S) = 563\). So \(P(\text{Speeding Ticket})=\frac{258}{563}\approx0.458\)

Step4: Part (c)

For conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here \(A\) is "having a speeding ticket" and \(B\) is "having a red car". \(P(A\cap B)=\frac{171}{563}\), \(P(B)=\frac{280}{563}\). Then \(P(\text{Speeding Ticket}|\text{Red Car})=\frac{171}{280}\approx0.611\)

Step5: Part (d)

For \(P(\text{Red Car}|\text{Speeding Ticket})\), \(P(A\cap B)=\frac{171}{563}\), \(P(B)=\frac{258}{563}\). Then \(P(\text{Red Car}|\text{Speeding Ticket})=\frac{171}{258}\approx0.663\)

Step6: Part (e)

For \(P(\text{Red Car and Speeding Ticket})\), \(n(A\cap B) = 171\) and \(n(S)=563\). So \(P(\text{Red Car and Speeding Ticket})=\frac{171}{563}\approx0.304\)

Step7: Part (f)

For \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). \(P(A)=\frac{280}{563}\), \(P(B)=\frac{258}{563}\), \(P(A\cap B)=\frac{171}{563}\). Then \(P(\text{Red Car or Speeding Ticket})=\frac{280 + 258-171}{563}=\frac{367}{563}\approx0.652\)

Answer:

a) \(0.497\)
b) \(0.458\)
c) \(0.611\)
d) \(0.663\)
e) \(0.304\)
f) \(0.652\)