He is the last guy you want to see in your rearview mirror w

He is the last guy you want to see in your rearview mirror when you are speeding down the highway, but research shows that a traffic ticket reduces a driver\'s chance of being involved in a fatal accident, at least for a few weeks. By age group, 14% of all drivers are younger than age 25, 56.6% are between ages 25 and 54, and 29.4% are 55 or older. Statistics show that 1.5% of the drivers younger than age 25, 2.3% of those ages 25 to 54, and 0.6% of those 55 or older will have an accident in the next month. What is the probability that a randomly identified driver will have an accident in the next month? (Give your answer correct to three decimal places.)

Of the parts produced by a particular machine, 1.3% are defective. If a random sample of 10 parts produced by this machine contains 2 or more defective parts, the machine is shut down for repairs. Find the probability that the machine will be shut down for repairs based on this sampling plan. (Give your answer correct to five decimal places.)

Solution

Let A1 shows the event that driver is younger than age 25, A2 shows the event that river is between ages 25 and 54 and A3 shows the event that driver is 55 or older. So we have following probabilties

P(A1)=0.14, P(A2)=0.566 and P(A3)=0.294

And let D shows the event that driver have an accident in teh next month.

P(D|A1)=0.015, P(D|A2)=0.023 and P(D|A3)=0.006

So the probability that a randomly identified driver will have an accident in the next month, P(D), is

P(D)=P(D|A1)P(A1)+P(D|A2)P(A2)+P(D|A3)P(A3)=0.015*0.14+0.023*0.566+0.006*0.294=0.016882

Hence, the required probability is 0.0169.