Ninety percent of all vehicles examined at a certain...

Ninety percent of all vehicles examined at a certain emissions inspection station pass the inspection. Assuming that successive vehicles pass or fail independently of one another, calculate the following probabilities. (Enter your answers to three decimal places.) (a) P(all of the next three vehicles inspected pass) (b) P(at least one of the next three inspected fails) (c) P(exactly one of the next three inspected passes) (d) P(at most one of the next three vehicles inspected passes) (e) Given that at least one of the next three vehicles passes inspection, what is the probability that all three pass (a conditional probability)? (Round your answer to three decimal places.)

Transcript

00:01 In this question, we are given at a certain admissions inspection station, 90 % of all vehicles examined will pass.

00:09 The next three vehicles are examined.

00:11 I'm going to let x be the number of vehicles out of these three that passes the inspection.

00:16 So end the number of trials is three, since there are three vehicles.

00:20 In each trial, we take a look at vehicle and see whether it passes the test or not.

00:25 So the three trials are independent, identical.

00:28 And in each trial there are only one or two possible outcomes, success or failure.

00:35 Success is the vehicle passes, failure is the vehicle does not pass.

00:38 And p probability of success in a single trial, that is probability of vehicle passes.

00:43 There will be 90 % of 0 .9 decimal and this remains constant.

00:47 So x follows the binomial distribution with n is 3, p 0 .9.

00:54 So probability of x equals to r, is the number of vehicles are on 3 that passes.

00:59 That will be 3 choose r 0 .9 to the power r and 1 minus 0 .9 is 0 .1 to the power of 3 minus r.

01:08 In part a, we want to find probability all 3 of them passes.

01:14 So we're looking at probability x equals to 3.

01:18 Just sub 3 into the r.

01:29 The answer is 0 .729.

01:33 In b, we want to find probability at least one of the 3 fails.

01:39 So x is the number of passing.

01:42 So number of failing will be 3 minus x.

01:45 So at least 1 will be greater equals to 1.

01:48 So we're looking at x less than equals to 2.

01:52 I'm going to take 1 minus the complement to this and that will be x greater than 2.

01:57 And for x greater than 2, there's only 1 case.

01:59 X is equal to 3.

02:02 So it will be 1 minus 0 .729...

Question

Please give Ace some feedback