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

d. Between 7 and 12, inclusive, believe that the minimum driving age should be 18.

Accepted Answer

Welcome back, everyone. A poll finds that 70% of college students own a laptop. In a random sample of 12 college students, what is the probability that between 5 and 9 inclusive on a laptop? So, for this problem, we have two possible outcomes, right? A college student may own a laptop or may not. So we're going to use a binomial probability distribution. We have a fixed number of trials, selections are independent, and the probability of success is 70%, right? It is constant. So what we want to do is simply identify the probability that our random variable X, which is the number of college students who own a laptop, is between 5 and 9 inclusive. So let's define that probability, and what we're going to do is simply define it as the probability of X being exactly 5. Plus the probability of X being 6. The probability of X being 7. Then we have the probability of X being 8, and finally the probability that X is 9. So, we have to find a total of 12345 probabilities and add them together. Remember the probability formula, the probability that X is equal to lowercase xis, and choose X multiplied by Ps the power of X, and multiplied by 1 minus P raises the power of N minus X. In this problem, and is the total number of college students, it is 12. For the values of X, we have those in our initial formula. And now the probability of success P is 70% or 0.70. So 1 minus P is going to be 0.30. That's the probability of failure. So we can now define the probability of. X being between 5 and 9 inclusive as follows using this formula, right? Starting with the probability that X is 5, we're going to choose. 12 Choose 5 to begin with, multiplied by. 0.70, raise to the power of 5, multiplied by 0.30, raise to the power of N minus X. So 12 minus 5 gives us 7. This is P X equals 5, and now we can continue this pattern. So the next probability we have is that X is equal to 6, that would be 12 to 6 multiplied by 0.70, raise to the power of 6, multiplied by 0.30 raise to the power of 6, 12 minus 6, right? Then we're going to add the probability that X is 7, so that's 12 to 7, multiplied by 0.70 to the power of 7, multiplied by 0.30 to the power of 5. Followed by 12, choose 8th. Multiplied by 0.70 to the power of 8, multiplied by 0. 30 to the power of 4, and finally we're going to add 12 to 9, multiplied by 0.70 to the power of 9 multiplied by 0.30 to the power of 12 minus 9 gives us 3. So now we can use a calculator to add all of those together and we're going to get 0.7377, which is our final answer for this problem. Let's label it and 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:
d. Between 7 and 12, inclusive, believe that the minimum driving age should be 18.

Key Concepts

Binomial Distribution

Probability of a Range of Outcomes

Random Sampling and Independence

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