Chapter 4. Q #47 f Q:The article “Should You Report That Fender- Bender?” (Consu

Question

Chapter 4. Q #47 f Q:The article “Should You Report That Fender- Bender?” (Consumer Reports, Sept. 2013: 15) reported that 7 in 10 auto accidents involve a single vehicle (the article recommended always reporting to the insurance company an accident involving multiple vehicles). Suppose 15 accidents are randomly selected.

f. What is the probability that exactly 4 involve a single vehicle and the other 11 involve multiple vehicles? It's asking the probability of X = 4 and Y = 11.

The answer key shows P(X=4) * P(Y=11).. My question is: since the sample size is 15, the probability of 4 single vehicle is the same as the p of 11 multiple vehicle, meaning if X=4, Y must be 11. thus why we still need to multiply the two Ps? I don't quite understand why the other 11 is independent of the number of accidents involved in a single vehicle.

Explanation / Answer

From the given information we want to find the probability that exactly 4 involve a single vehicle and the other 11 involve multiple vehicles. So we need to find only P( X = 4) and don't need to multiply P(Y = 11).

Because there are 15 random samples and so that P( X = 4) means  the probability that exactly 4 involve a single vehicle and the other 11 involve multiple vehicles.

By using binomial formula with parameter n = 15, and p = 0.7

P( X = 4) = 15C4 *(.7)^4 *(0.3)^11 = 0.000581

Note that if we find P( Y = 11) using n = 15 and p = 0.3 then we get the same answer.

There may be printing mistake in the answer Key.

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