Find the indicated probabilities. If convenient, use...

Find the indicated probabilities. If convenient, use technology or Table 2 in Appendix B to find the probabilities. Safety Recall Sixty-eight percent of adults would still consider a car brand despite product/safety recalls. You randomly select 20 adults. Find the probability that the number of adults who would still consider a car brand despite product/safety recalls is (a) exactly one, (b) more than one, and (c) at most one.

Find the indicated probabilities. If convenient, use technology or Table 2 in Appendix B to find the probabilities.
Safety Recall Sixty-eight percent of adults would still consider a car brand despite product/safety recalls. You randomly select 20 adults. Find the probability that the number of adults who would still consider a car brand despite product/safety recalls is (a) exactly one, (b) more than one, and (c) at most one.

Key Concepts

Cumulative Probability and Complement Rule

Cumulative probability involves summing the probabilities of several outcomes to cover a range of interest, such as 'at most' or 'more than' a certain number of successes. The complement rule is a useful tool in these cases; for example, calculating the probability of more than one success can be done by subtracting the probabilities of zero and one success from one, thereby simplifying the computation.

Binomial Distribution

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. It applies when each trial results in one of two outcomes (often labeled 'success' and 'failure'), and the trials are independent of each other.

Binomial Probability Formula

This formula, P(X = k) = C(n,k) * p^k * (1-p)^(n-k), is used to calculate the probability of achieving exactly k successes in n independent trials where each trial has a success probability of p. In the context of the problem, it helps determine the likelihood of a specific number of adults considering the car brand.

Transcript

00:01 So this problem is going to be a binomial probability problem because it meets certain characteristics.

00:16 As we read through it, you're going to find that there are a fixed number of trials.

00:24 You're going to find that the trials are independent.

00:36 And you're going to see that there are only two outcomes.

00:43 And we usually define those outcomes as success and failure.

00:53 And whenever you're working with binomial probabilities, there are four variables that you are going to work with.

01:00 The first one is n, which represents the number of trials.

01:08 X, which would be the possible outcomes, or the counts of how many successes.

01:22 P, which is the probability of success.

01:24 And q, which is the probability of failure.

01:39 And there's one formula that we are going to use to govern these types of problems, and that is going to be the probability that an event occurs is equal to ncx, so our combination notation, multiplied by p raised to the x power, multiplied by q raised to the n minus x power.

02:03 So let's read through the problem and gather the values of our variables.

02:10 So as we read through, it says that there are 68 % of adults would still consider a car brand despite the product recalls.

02:21 So that means we have a p value of 0 .68.

02:26 And if 68 % would still consider the cars regardless of safety recalls, that means there are 32 % that would not.

02:39 You're randomly going to select 20 adults, so our n is going to be 20 throughout this problem, and then we get into the actual questions.

02:50 And it says, find the probability that the number of adults who would still consider a car brand, despite the safety recalls is, and we're going to do part a, and it's exactly one.

03:02 So we are talking about x being one in this case.

03:07 So therefore, we're going to say the probability that x equals 1 will be 20c1, multiplied by 0 .68 raised to the first power, multiplied by 0 .32 raised to the 20 minus 1 power.

03:34 And the probability of finding exactly 1 out of those 20, when you type this into the calculator, gives you an answer in scientific notation.

03:45 So you're going to see 5 .387 -515, and in most calculators you're going to see e negative, and it's because our answer is super small that we end up putting it into scientific notation.

04:04 And what it means is going to be that you're going to move the decimal point nine places to the left and get this probability.

04:21 0 .00000, 0 .00 -0 -0 -0 -0 -3513 -87515.

04:29 So it's very, very, very, highly unlikely.

04:32 That when you select 20 random adults, only one would consider the car regardless of the product safety recalls.

04:45 Now let's look at part b.

04:48 And in doing part b, we're going to have to think a little bit differently.

04:53 So in part b, you want the probability that the number of adults would be more than one.

05:04 So that means we are focused on the x values of 2 or 3 or 4 or 5, 6, 7, 8, all the way up to the 20...

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