Problem 40 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The probability is approximately 0.00596.

Step 1: Understand the Problem

We have a total of 14,000 college student smokers. We need to find the probability that no more than 3 out of 10 randomly selected student smokers smoke less than 1 cigarette per day. The percent given is 0.45, which represents the probability for a single student smoker.

Step 2: Use the Binomial Distribution

The situation can be modeled using the binomial distribution since we are dealing with a fixed number of trials (10 students), each with two possible outcomes, and constant probability. Set the number of trials as \(n = 10\), the probability of success (smoking less than 1 cigarette) as \(p = 0.45\), and the number of successes as \(k\). We need to find \(P(X \leq 3)\).

Step 3: Calculate Individual Probabilities

Calculate the binomial probability for each possible value of \(k\) (0 through 3). The formula for a binomial probability is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \(\binom{n}{k}\) is the binomial coefficient. Calculate these probabilities for \(k = 0, 1, 2, 3\).

Step 4: Sum the Probabilities for the Desired Outcome

Sum the probabilities calculated in Step 3 to get \(P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)\). This will give us the probability that no more than 3 students smoke less than 1 cigarette per day.

Step 5: Apply the Binomial Formula

Calculate:\(P(X = 0) = \binom{10}{0} (0.45)^0 (0.55)^{10} = 0.000034980460\)\(P(X = 1) = \binom{10}{1} (0.45)^1 (0.55)^9 = 0.000286548707\)\(P(X = 2) = \binom{10}{2} (0.45)^2 (0.55)^8 = 0.001345234308\)\(P(X = 3) = \binom{10}{3} (0.45)^3 (0.55)^7 = 0.004296160735\)

Step 6: Calculate the Total Probability

Add up all the probabilities for \(k = 0\) to \(k = 3\): \[P(X \leq 3) = 0.00003498 + 0.00028655 + 0.00134523 + 0.00429616 = 0.0059630238\]

Key Concepts

Probability Calculation

Probability is the measure of the likelihood that a particular event will occur. In the context of this exercise, we are calculating the probability of a specific outcome related to cigarette smoking habits among college students. To do this, we use the binomial distribution, which is handy when you have:

• A fixed number of trials or experiments, in this case, selecting 10 students.
• Two possible outcomes, such as whether a student smokes less than 1 cigarette or not.
• A constant probability of success on each trial, here, a student smokes less than 1 cigarette with probability 0.45.

The core of this calculation involves the binomial probability formula:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Here:- \( n \) is the number of trials, which is 10.- \( k \) is the number of students we expect to smoke less than 1 cigarette, ranging from 0 to 3 in this problem.- \( p \) is the given probability of a student smoking less than 1 cigarette, 0.45 in this case.
By calculating the probability for each value of \( k \) (0, 1, 2, and 3), and then summing them up, we find \( P(X \leq 3) \), the probability that no more than three selected students smoke less than 1 cigarette per day.

Cigarette Smoking Survey

Surveys are crucial for collecting data about behaviors, such as cigarette smoking among college students. The data from the survey helps us identify the distribution of smoking habits among students. In this case, the survey provides the following information:

• 45% of students smoke less than 1 cigarette per day.
• 24% smoke 1 to 9 cigarettes per day.
• 20% smoke 10 to 19 cigarettes per day.
• 11% smoke a pack of 20 or more cigarettes daily.

This data is given in decimal format, making it convenient for probability calculations, as the decimals directly represent probabilities. The probability for a single student's smoking habit becomes a crucial input when applying the binomial distribution.
For example, the 0.45 probability is used extensively in the calculations, reflecting the habits of students who smoke less than 1 cigarette daily. This approach underlines how statistical data from surveys aid in deeper analysis through probability models.

College Student Statistics

Statistics about college students, such as their smoking habits, offer valuable insights into health and behavior trends on campuses. Knowing that out of a large population, a particular percentage engages in a specific behavior, allows for targeted public health approaches and educational programs.
The survey results show smoking trends among college students, useful for campus health services in creating campaigns for smoking cessation or prevention programs. The breakdown is a window into how many students may need support to quit smoking and those at risk of developing habits leading to potential health issues.
These statistics, derived from surveys, assist not only in understanding the behavior but also in modeling different scenarios using probability and statistics. By choosing random samples, like the 10 students, and applying mathematical models, we derive probabilities that inform predictions and, ultimately, decisions aimed at improving student health and wellbeing.