Problem 33 - Algebra and Trigonometry

Question

Smokers and Nonsmokers The participants at a mathematics conference are housed dormitory-style, five to a room. Because of an oversight, conference organizers forgot to ask whether the participants are smokers. In fact, it turns out that 30\(\%\) are smokers. Find the probability that Fred, a nonsmoking conference participant, will be housed with (a) Exactly one smoker (b) One or more smokers

Step-by-Step Solution

Answer

(a) 0.4116 (b) 0.7599

Step 1: Understanding the Problem

We know that each room houses 5 participants. Fred is one of these participants and is a nonsmoker. This means there are 4 other occupants in the same room. The probability that any given participant is a smoker is 30\(\%\). We need to find the probability that Fred is housed with exactly one smoker and at least one smoker.

Step 2: Determining Probability of "Exactly One Smoker"

We will use the binomial probability formula given by \( P(X = k) = \binom{n}{k} p^k(1-p)^{n-k} \), where \( n \) is the number of trials, \( k \) is the number of successful trials, and \( p \) is the probability of success. Here, \( n = 4 \) (since Fred's presence means we're only considering the other 4), \( k = 1 \), and \( p = 0.3 \).\[P(X = 1) = \binom{4}{1} (0.3)^1(0.7)^3\]\[= 4 \times 0.3 \times 0.343\]\[= 0.4116\]

Step 3: Calculating Probability of "One or More Smokers"

To find the probability of housing with at least one smoker, we approach it by finding the complement probability: the probability of no smokers in the room.The probability of all 4 being nonsmokers is \( (0.7)^4 \).\[P(\text{no smokers}) = (0.7)^4 = 0.2401\]The probability of having one or more smokers is the complement of this:\[P(\text{one or more smokers}) = 1 - 0.2401 = 0.7599\]

Key Concepts

Understanding the Binomial Probability Formula

Probability theory is all about determining how likely an event is to occur. The binomial probability formula is a crucial part of this field, especially when dealing with events that have two possible outcomes. Imagine you are flipping a coin: heads or tails. Similarly, in the context of our problem, each conference participant is either a smoker or a nonsmoker.

The binomial probability formula helps us determine the probability of having a fixed number of successes in a set number of trials. Here's what the formula looks like:

• The probability of exactly \( k \) successes in \( n \) trials is expressed as \( P(X = k) = \binom{n}{k} p^k(1-p)^{n-k} \).
• \( n \) denotes the number of trials or opportunities (e.g., the other four people besides Fred in his room).
• \( k \) represents the number of successful outcomes we're looking at (like having exactly one smoker among Fred's roommates).
• \( p \) is the probability of one trial resulting in success (30\% chance that a participant is a smoker in this case).

The formula involves combinations because it considers the different ways to choose \( k \) successes out of \( n \), which is where the combination notation, denoted by \( \binom{n}{k} \), comes in. Understanding this is key to applying the formula correctly.

The Role of Combinatorial Mathematics in Probability

Combinatorial mathematics provides the framework for studying different arrangements and selections, known respectively as permutations and combinations. This branch of mathematics is central to probability because it helps us figure out how many ways events can occur.

For our problem, combinatorics explains how to calculate the number of ways to select smokers from the potential roommates.

• The notation \( \binom{n}{k} \) reads as \( "n \text{ choose } k" \) and corresponds to the number of ways to choose \( k \) successes (smokers) out of \( n \) total trials (Fred's other roommates).
• In the example, \( \binom{4}{1} \) computes the number of ways one smoker can be selected from four potential roommates, resulting in 4 different combinations.
• This computation allows us to apply the binomial formula by determining the number of possible favorable outcomes versus all outcomes.

The power of combinatorial mathematics is in breaking down complex probability problems into manageable parts, making it easier to derive meaningful and accurate results.

Utilizing Statistical Analysis for Predictions

Statistical analysis allows us to make informed predictions based on data and probability models. For instance, in our problem, statistical analysis helps us estimate Fred's chances of rooming with smokers.

Here's how statistical analysis ties into our solution:

• It provides a systematic approach to calculating the probability of both exactly one smoker and at least one smoker being in Fred's room using the foundational principles like complement probability.
• By understanding the concept of complements, we compute the probability of one or more smokers by subtracting the probability of no smokers from 1.
• This method is rooted in analytical thinking common in statistical contexts, where calculations often help uncover subtler truths behind data observations.
• We can infer broader conclusions by estimating real-world probabilities using smaller sample spaces, making statistical analysis a vital tool for decision-making.

Through statistical analysis, Fred and other stakeholders can confidently anticipate scenarios, helping confer organizational strategies for future events.