Driving AgeAccording to a Gallup poll, 60% of U.S. women 18 ye...

Question

Driving AgeAccording to a Gallup poll, 60% of U.S. women 18 years old or older stated that the minimum driving age should be 18. In a random sample of 15 U.S. women 18 years old or older, find the probability that:

b. Fewer than 5 believe that the minimum driving age should be 18

b. Fewer than 5 believe that the minimum driving age should be 18

Accepted Answer

Welcome back, everyone. Suppose 55% of adults support a new recycling program. In a random sample of 10 adults, what is the probability that fewer than 2 support the program? A 0.0062, B 0.0045, C 0.0128, and D 0.0214. So for this problem, we're going to use the binomial probability formula because we have two outcomes, a fixed number of trials, independent trials, and therefore we can use the probability formula, the probability of a random variable X being equal to lowercase x is equal to n choose x multiplied by P to the power of X multiplied by 1 minus P. Raise the power of N minus X, right? Now, our probability of success P is 55%, which is 0.55. Our N, which is the number of trials, corresponds to 10 adults. This is how we can interpret it, right? And essentially what we want to do is identify the probability that our random variable X, which is the number of adults, right? That support a new recycling program is fewer than 2, so we're looking for the probability of X being less than 2. Now, how do we identify that probability? Well, essentially the probability that X is less than 2. Would be the probability of X being exactly one plus the probability. That X is exactly 0. Since X can only be an integer, we start at 0, then we get 1, and that's it. So we have to find two probabilities and add them together. So we can show that the probability. Of X being less than 2 is equal to. Starting with the probability of X equals 1, we're going to take N of 10, so we take them, choose 1 because now X is equal to 1. We're going to multiply that by the probability of success 0.55. Raise the power of X. So if X is one we're raising it's the power of 1, and multiplying by 1 minus 0.55, raises the power of. N minus X. So we take 1 minus 1, and then we're going to add the probability of X being 0. So we're adding some choose 0. Multiplied by 0.55, raise the power of X. So in this case that's 0. Multiplied by 1 minus 0.55, raise the power of N minus X. So 10 minus 0. Well then, so now what we can do is simplify and then use a calculator to get the result, right? So starting with the first probability. And choose one is some. We multiply that by 0.55 and multiply by 0.45, raise the power of 9. Then We are going to add 10 to 0, so that's basically 1, multiplied by 0.55 to the power of 0, that's 1, and then we're multiplying by 0.45 to the power of 10. Well then, so now that we have simplified, we can add these numbers and we're going to get 0.0045. Which corresponds to the answer choice B. Thank you for watching.

Driving AgeAccording to a Gallup poll, 60% of U.S. women 18 years old or older stated that the minimum driving age should be 18. In a random sample of 15 U.S. women 18 years old or older, find the probability that:

b. Fewer than 5 believe that the minimum driving age should be 18

Key Concepts

Binomial Distribution

Probability of 'Fewer than' Events

Random Sampling and Independence

Watch next

Driving AgeAccording to a Gallup poll, 60% of U.S. women 18 years old or older stated that the minimum driving age should be 18. In a random sample of 15 U.S. women 18 years old or older, find the probability that: b. Fewer than 5 believe that the minimum driving age should be 18

Key Concepts

Binomial Distribution

Probability of 'Fewer than' Events

Random Sampling and Independence

Watch next

Driving AgeAccording to a Gallup poll, 60% of U.S. women 18 years old or older stated that the minimum driving age should be 18. In a random sample of 15 U.S. women 18 years old or older, find the probability that:

b. Fewer than 5 believe that the minimum driving age should be 18