91Ó°ÊÓ

Political affiliation. Suppose that in a large population, \(51 \%\) identify as Democrat. A researcher takes a random sample of 3 people. (a) Use the binomial model to calculate the probability that two of them identify as Democrat. (b) Write out all possible orderings of 3 people, 2 of whom identify as Democrat. Use these scenarios to calculate the same probability from part (a) but using the Addition Rule for disjoint events. Confirm that your answers from parts (a) and (b) match. (c) If we wanted to calculate the probability that a random sample of 8 people will have 3 that identify as Democrat, briefly describe why the approach from part (b) would be more tedious than the approach from part (a).

Step by step solution

01

Defining the Binomial Model

The binomial model describes the probability of obtaining a fixed number of successful trials in a fixed number of independent experiments. For this exercise, identify a 'success' as a person identifying as a Democrat and use the success probability ( p = 0.51 ) for the binomial distribution parameters. We have n = 3 trials (people), and we need to find the probability of k = 2 successes (identifying as Democrats).

02

Apply Binomial Probability Formula for Part (a)

The probability of exactly kevents happening in ntrials can be found using the binomial probability formula:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Here, \( p = 0.51 \), \( n = 3 \), and \( k = 2 \). Substitute these values:\[ P(X = 2) = \binom{3}{2} (0.51)^2 (0.49)^1 \]Calculate this step-by-step:\[ \binom{3}{2} = 3 \]\[ (0.51)^2 = 0.2601 \]\[ (0.49)^1 = 0.49 \]Multiply these results:\[ 3 \times 0.2601 \times 0.49 = 0.382347 \]

03

Listing All Possible Orderings for Part (b)

List all possible outcomes for selecting 3 people where exactly 2 identify as Democrats:1. D, D, R2. D, R, D3. R, D, DWhere 'D' stands for Democrat and 'R' for not Democrat (Republican or other). Each ordering has these probabilities calculated as \[ 0.51 \times 0.51 \times 0.49 = 0.126847 \]

04

Summing Probabilities for Disjoint Events

Using the Addition Rule for disjoint events, calculate the total probability of exactly two Democrats by adding up the probabilities of each ordering:\[ P(DDR) + P(DRD) + P(RDD) = 0.126847 + 0.126847 + 0.126847 \]\[ = 0.380541 \]This should be approximately equal to the result from Step 2 due to rounding in calculations.

05

Explanation for Part (c)

Calculating the probability for 3 out of 8 identifying as Democrat using the ordering method is tedious because of the sheer number of combinations. With \( n = 8 \) and \( k = 3 \), the number of combinations is \( \binom{8}{3} = 56 \), meaning 56 different sequences must be considered using the Addition Rule. The binomial model directly applies the formula without listing combinations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability

Probability is a fundamental concept in statistics, useful for predicting the likelihood of an event. When dealing with events such as a person identifying as a Democrat, probability helps us understand how often that event might occur. In our case, the probability (denoted as \( p \)) is the chance that a randomly selected person identifies as a Democrat, which is 51%, or 0.51 in decimal form.

Probability is calculated in terms of outcomes. For example, if you randomly choose one person, the chance they will be a Democrat is 0.51, and the chance they will not be is 0.49 (which is 1 minus 0.51). Probability values always range from 0 (impossible event) to 1 (certain event).

• If \( p = 0.5 \), each outcome is equally likely.
• If \( p > 0.5 \), the event is more likely to happen.
• If \( p < 0.5 \), the event is less likely to happen.

Disjoint Events

Disjoint events, also known as mutually exclusive events, cannot happen at the same time. For example, in a coin flip, landing on heads and tails at the same time is impossible. In the context of our binomial distribution problem, each specific ordering of Democrats and non-Democrats is a disjoint event.

This concept is crucial when calculating probabilities of occurring events since each possible outcome needs to be independent of the others. For this exercise, when listing outcomes such as (D, D, R), (D, R, D), and (R, D, D), these are disjoint events. Each sequence is unique and does not overlap with the others.

• Disjoint events help us see that different outcomes don't interfere with each other.
• This is useful when adding probabilities to find the overall likelihood of a compound event.

Addition Rule

The Addition Rule is a principle that helps us find the probability of one event or another occurring when events are disjoint. In simple terms, to find the probability of any of the disjoint events happening, sum their individual probabilities.

For our problem with three people, the probability of exactly two identifying as Democrats can be calculated by adding the probabilities of all disjoint outcomes: (D, D, R), (D, R, D), and (R, D, D). Each outcome has its probability calculated and then summed to find the total probability.

• For disjoint events, like our scenarios here, \( P(A \text{ or } B) = P(A) + P(B) \).
• Make sure events are disjoint; otherwise, this rule doesn't apply.

Combinatorics

Combinatorics deals with counting and arranging objects. It’s crucial for determining how many ways an event can happen, particularly in complex scenarios with multiple trials, like our problem with 8 people.

In a scenario where you want 3 out of 8 people to identify as Democrats, you'd use a combination calculation. The number of ways to choose \( k \) successes (Democrats) from \( n \) trials (people) is given by the combination formula \( \binom{n}{k} \).

This explains why using binomial models is efficient: you compute \( \binom{8}{3} = 56 \), representing 56 possible combinations of 3 Democrats among 8 people, rather than writing each sequence.

• Using combinatorics streamlines complex probability problems.
• Combinatorics is used in the binomial probability formula.