The table gives the results of a survey of 14,000 college...

The table gives the results of a survey of $14,000$ college students who were cigarette smokers in a recent year. $$\begin{array}{|l|c|} \hline \begin{array}{l} \text { Number of Cigarettes } \\ \text { per Day } \end{array} & \begin{array}{c} \text { Percent } \\ \text { (as a decimal) } \end{array} \\ \hline \text { Less than } 1 & 0.45 \\ 1 \text { to } 9 & 0.24 \\ 10 \text { to } 19 & 0.20 \\ \text { A pack of } 20 \text { or more } & 0.11 \end{array}$$ Using the percents as probabilinies, approximate the probability that, out of 10 of these shudent smokers selected at random, the following were true. No more than 3 smoked less than 1 cigarette per day.

The table gives the results of a survey of $14,000$ college students who were cigarette smokers in a recent year.
$$\begin{array}{|l|c|} \hline \begin{array}{l} \text { Number of Cigarettes } \\ \text { per Day } \end{array} & \begin{array}{c} \text { Percent } \\ \text { (as a decimal) } \end{array} \\ \hline \text { Less than } 1 & 0.45 \\ 1 \text { to } 9 & 0.24 \\ 10 \text { to } 19 & 0.20 \\ \text { A pack of } 20 \text { or more } & 0.11 \end{array}$$
Using the percents as probabilinies, approximate the probability that, out of 10 of these shudent smokers selected at random, the following were true.
No more than 3 smoked less than 1 cigarette per day.

Key Concepts

Binomial Distribution

This is a probability distribution that models the number of successes in a fixed number of independent trials, where each trial has two possible outcomes and a constant probability of success. It is used to calculate the probability of obtaining a specific number of successes out of a total number of trials.

Probability Mass Function

The probability mass function (PMF) for the binomial distribution provides a formula to determine the probability of obtaining exactly a given number of successes. The PMF includes a binomial coefficient to account for the number of ways the successes can occur among the trials, multiplied by the probabilities of success and failure raised to the appropriate powers.

Combinatorics

This field of mathematics deals with counting, particularly in determining how many ways events can occur. In the context of the binomial distribution, combinatorial concepts like 'n choose k' are used to calculate the number of different ways a certain number of successes can be arranged among the trials.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a certain threshold. When working with the binomial distribution, it involves summing the probabilities of all outcomes up to a specified number of successes.

Probability Using Percentages

When percentages are provided, they can be directly converted into probabilities by expressing them as decimals. This conversion is essential for applying probabilistic models like the binomial distribution, where the success probability must be between 0 and 1.

Transcript

00:01 To solve this problem, we're going to need to reference the chart that shows the percent as a decimal of college students who smoke the given number of cigarettes per day.

00:11 And given that chart, i want to know, if i take a random sample of 10 of these student smokers, what's the probability that no more than three of them smoked less than one cigarette per day? well, there are four different scenarios that match my criteria.

00:30 If i want no more than three to have smoked less than one cigarette per day, i could have three people that fit that criteria, two people, one person, or no people that have smoked less than one cigarette per day.

00:50 And this is an or.

00:51 I could have three or two, or one, or zero.

00:55 In probability, when we use or, we're going to add.

00:59 So i will will individually find the probability of each of these cases happening and i will add up those probabilities, that will give me my overall probability of no more than three who smoked less than one cigarette per day.

01:13 Well, let's take a look at the first one.

01:16 Three people match my criteria.

01:18 I've got a group of 10 people.

01:21 And since i want three people to match, i'm going to start with a combination of 10 things taken three at a time.

01:27 I'm going to multiply that by the probability of somebody matching my criteria raised to the third power.

01:35 And since i want seven people who don't match my criteria, i'm going to take that probability and raise it to the seventh power.

01:43 So what's the probability that somebody smokes less than one cigarette per day? well, according to that chart, that probability is 0 .45...

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