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People Who Have Some College Education Fifty-three percent of all persons in the U.S. population haveat least some college education. Choose 10 persons atrandom. Find the probability that $$\begin{array}{l}{\text { a. Exactly one-half have some college education }} \\ {b \text { . At least } 5 \text { do not have any college education }} \\ {\text { c. Fewer than } 5 \text { have some college education }}\end{array}$$

a. 0.242b. 0.547c. 0.306

Intro Stats / AP Statistics

Chapter 5

Discrete Probability Distributions

Section 3

The Binomial Distribution

Sampling and Data

Probability Topics

Probability and Counting Rules

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53% of people have a college education branch, use 10 people at random, their value of and is 10. And our probability of success is 0.53 for this one success, meaning they have a college education. First question says, What is the probability that exactly half of the people surveyed have a college education? So we're looking for the probability that X is equal to five. Since there's 10 people surveyed, then half of that would be five. So using the binomial distribution formula, we do 10 factorial, divided by five factorial times five factorial times 0.53 to the fifth power times, 0.47 to the fifth power. We got 0.47 by doing one minus 10.53 That equals 0.47 This would be the probability that they don't have some sort of college education. So when we do 10 factorial divided by five factorial times five factorial, you get 252 times 0.53 to the fifth power times 0.47 to the fifth. Power is equal to 0.242 or 24.2%. Remember, you can do this in the cat litter by hitting second bars you will look for by gnome. Pdf is the same equals 25 You do 10 0.53 and five and you would get 0.242 What if the question says at least five do not get a college education? So we're looking for What is the probability that X is greater than or equal to five? And this one we're looking for do not have some sort of college education. So therefore, you wanna look at the the other probability of 0.47 because 0.53 says they have some sort of college education. 0.47 says they have no college education. We're gonna dio one minus by gnome CDF our sample sizes 10. Our probability for this one is 0.47 and we are looking for greater than or equal to five. So we do less than or equal to four. This probability will tell us less than or equal to four, and we subtract that from one to get all the numbers that are greater than then four. When I do that, you still use the same function in the Cacador second bars, you'll see by gnome CDF and you get 0.547 or 54.7%. You could do this by hand, but you're gonna have to find the probability that X equals 56789 10 And add them all together and you get the same answer. What is the probability Probability that fewer than five have a college education. To do that, you're looking for the probability that X is less than five. So I can do the probability of X equals zero plus the probability of X equals one, plus the probability of X equals two all the way up to the probability the X equals four. I don't include five in this one because it just says fewer than five have a college education. If I could do all of that by hand, using the binomial formula or I can use the Bynum, he will live distribution function. And I would you 10 0.53 and four, cause I'm looking for the sum of all the probabilities from 0 to 4. I do that in the keg bitter. I get 0.306 which is 30.6%

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