According to the USA Today Snapshot "Knowing drug addicts,"...

According to the USA Today Snapshot "Knowing drug addicts," $45 \%$ of Americans know somebody who became addicted to a drug other than alcohol. Assuming this to be true, what is the probability of the following? a. Exactly 3 people of a random sample of 5 know someone who became addicted. Calculate the value. b. Exactly 7 people of a random sample of 15 know someone who became addicted. Estimate using Table 2 in Appendix B. c. At least 7 people of a random sample of 15 know someone who became addicted. Estimate using Table 2. d. No more than 7 people of a random sample of 15 know someone who became addicted. Estimate using Table 2.

According to the USA Today Snapshot "Knowing drug addicts," $45 \%$ of Americans know somebody who became addicted to a drug other than alcohol. Assuming this to be true, what is the probability of the following?
a. Exactly 3 people of a random sample of 5 know someone who became addicted. Calculate the value.
b. Exactly 7 people of a random sample of 15 know someone who became addicted. Estimate using Table 2 in Appendix B.
c. At least 7 people of a random sample of 15 know someone who became addicted. Estimate using Table 2.
d. No more than 7 people of a random sample of 15 know someone who became addicted. Estimate using Table 2.

Key Concepts

Bernoulli Trials

Bernoulli trials are experiments or processes that result in a binary outcome, typically labeled as success or failure. Each trial is independent and the probability of success remains constant, forming the basis for the binomial model.

Binomial Distribution

This concept models a fixed number of independent trials, where each trial has two outcomes—success or failure—with a constant probability of success. It is used to calculate the probabilities of obtaining a specific number of successes in a set number of trials.

Probability Mass Function (PMF)

The PMF for the binomial distribution gives the probability of obtaining exactly k successes in n independent trials. It is calculated using the formula P(X = k) = (n choose k) * p^k * (1 ? p)^(n ? k), where p is the probability of success in each trial.

Cumulative Probability

Cumulative probability is the sum of probabilities for a range of outcomes. In the context of the binomial distribution, it is used to determine the probability of obtaining at least or no more than a given number of successes by summing the individual probabilities for each outcome in the range.

Transcript

00:01 So 45 % of people know someone who is addicted to a drug other than alcohol.

00:07 So this is a binomial random variable because there are two options.

00:14 You're either addicted to this drug or you're not.

00:17 So that means we can use this equation to find probabilities because this gives us probabilities of binomial random variables.

00:27 And i think you should have this written down.

00:31 Now for the first one, find probability of free from a sample of five.

00:39 So the free pieces of information we need are the five, the free, and then this 45 up here.

00:48 Five is our sample, that's n.

00:51 X is going to be free.

00:53 That's the number we're looking for.

00:55 And our probability is 45%.

00:58 Now we're going to convert this to a decimal.

01:01 Though, so it's going to be 0 .45.

01:06 So to find the probability of free, that'll be equal to the sample 5 choose free times the probability, so 0 .45 cubed to the 1 minus 0 .45 ,000, multiplied by 1 minus 0 .45, raised to the 5 minus free power.

01:30 And this is going to give us 0 .276.

01:37 Now we have a sample of 15 people, and we need to find the probability that seven of them are addicted to the drug, but you need to use the table in the back of the book.

01:49 In appendix b, table 2, it gives you a table that looks something like this.

01:55 I cannot post pictures directly from the book, so i recreated some of it.

02:00 So you're going to scroll down to the n equals 15 section.

02:07 And because our probability is 45%, you're going to be in between the 0 .40 and the 0 .50.

02:17 So i wrote out some of the relevant section here.

02:21 So to find probability of 7, well, we go to 7...

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