91Ó°ÊÓ

Let \(X_{1}, X_{2}, X_{3}, X_{4}, X_{5}\), and \(X_{6}\) denote the numbers of blue, brown, green, orange, red, and yellow M\&M candies, respectively, in a sample of size \(n\). Then these \(X_{i}\) s have a multinomial distribution. According to the M\&M web site, the color proportions are \(p_{1}=.24, p_{2}=.13, p_{3}=.16, p_{4}=\) \(.20, p_{5}=.13\), and \(p_{6}=.14\). a. If \(n=12\), what is the probability that there are exactly two M\&Ms of each color? b. For \(n=20\), what is the probability that there are at most five orange candies? [Hint: Think of an orange candy as a success and any other color as a failure.] c. In a sample of \(20 \mathrm{M} \& \mathrm{Ms}\), what is the probability that the number of candies that are blue, green, or orange is at least 10 ?

Step by step solution

01

Understanding the Multinomial Distribution

The multinomial distribution is an extension of the binomial distribution. It models the probability of counts for multiple categories that are mutually exclusive. In the problem, we have six different colors, and we know their proportions and the total number of candies. We need to use this information to find the probabilities of specific combinations of candies.

02

Part (a): Calculating Probability for Exact Count with n=12

For part a, we need the probability of getting exactly two of each color when we have 12 candies. Use the multinomial probability formula:\[ P(X_1 = 2, X_2 = 2, X_3 = 2, X_4 = 2, X_5 = 2, X_6 = 2) = \frac{12!}{2!2!2!2!2!2!} \times 0.24^2 \times 0.13^2 \times 0.16^2 \times 0.20^2 \times 0.13^2 \times 0.14^2 \].Calculate this to find the exact probability.

03

Solving Part (a) with Calculations

Calculate each part:- The coefficient \( \frac{12!}{2!2!2!2!2!2!} \) simplifies to 10395.- Calculate the probabilities \( 0.24^2 \times 0.13^2 \times 0.16^2 \times 0.20^2 \times 0.13^2 \times 0.14^2 = 1.9959168 \times 10^{-6}\).- Multiply: \( 10395 \times 1.9959168 \times 10^{-6} \approx 0.020728 \). So, the probability is approximately 0.0207.

04

Part (b): Using Binomial Distribution for Orange Candies

For part b, consider the orange candies as 'successes' in the binomial distribution with \( n = 20 \) and success probability \( p_4 = 0.20 \). We seek the probability of \( X_4 \leq 5 \).Use the binomial probability formula to sum probabilities from 0 to 5:\[P(X_4 \leq 5) = \sum_{k=0}^{5} \binom{20}{k} 0.20^k (0.80)^{20-k}\].Calculate this sum using a binomial cumulative distribution function.

05

Solving Part (b) with Calculations

Calculate each term from the binomial sum:- Use a calculator or statistical software to find \( P(X_4 \leq 5) \), where \( p = 0.20 \) and \( n = 20 \), resulting in approximately 0.5881.

06

Part (c): Calculating Probability for At Least 10 Blue, Green or Orange

For part c, combine the probabilities for blue, green, and orange candies. The combined probability for these is \( p_1 + p_3 + p_4 = 0.24 + 0.16 + 0.20 = 0.60 \).Calculate the probability of getting at least 10 candies of these combined categories using a binomial distribution where \( n = 20 \) and \( p = 0.60 \):\[P(X \geq 10) = 1 - P(X < 10) = 1 - \sum_{k=0}^{9} \binom{20}{k} 0.60^k (0.40)^{20-k}\].

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Key Concepts

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

Binomial Distribution

The binomial distribution is a fundamental concept in probability theory. It models situations where a fixed number of trials are conducted, each with two possible outcomes: success or failure. This distribution assumes that each trial is independent and that the probability of success, denoted by \( p \), remains constant throughout. When you think about "success," it can be any event of interest, such as picking an orange-colored M&M.

• The number of trials is represented by \( n \).
• The probability of success in each trial is \( p \).
• A classic example could be flipping a coin where getting heads is considered a success.

For the orange M&Ms' scenario, we view picking an orange candy as a success with \( p = 0.20 \). By using the binomial distribution formula, we calculate the probability of a range of outcomes.

Probability Calculation

Calculating probability often involves straightforward steps, but can be complex when multiple events or categories are involved. This process is at the heart of determining how likely it is for an event or a combination of events to occur.In Part (a) of the problem, we use the multinomial distribution to find the probability of obtaining two M&Ms of each color when \( n = 12 \). This requires us to:

• Compute a coefficient based on the total number of candies and the desired count for each color.
• Multiply this coefficient by the product of raising each color's proportion to the power of its count.

For parts where the binomial distribution is suitable, as in Part (b) and (c) with orange M&Ms, the calculation involves summing probabilities of individual outcomes, or using cumulative distribution functions for efficiency.

Combinatorial Probability

Combinatorial probability involves determining the likelihood of various combinations of outcomes. It's important for problems that have multiple categories or require specific arrangements of outcomes.Key elements include factorials and combination formulas. For example:

• The formula \( \binom{n}{k} \) represents choosing \( k \) successes in \( n \) trials and is fundamental in calculating binomial probabilities.
• For multinomial distributions, the multiple outcomes and categories add layers of complexity through factorial divisions and extensive product calculations.
• In Part (a) of the original problem, the multinomial coefficient \( \frac{12!}{2!2!2!2!2!2!} \) is calculated to determine the number of ways to arrange the candies.

Understanding these concepts helps in tackling increasingly complex probability calculations.

Color Probability

Color probability specifically examines events related to the likelihood of selecting different colors from a set. In the context of M&Ms, each color has a known probability or proportion in the mix.To solve exercises involving color probability, consider:

• The proportion of each color, provided directly by the problem or empirical data, as critical inputs to computation.
• Each color acts as a distinct category or event with its unique probability, which is taken into account simultaneously, especially in multinomial distributions.
• In scenarios where multiple colors are combined, such as blue, green, or orange M&Ms in Part (c), their probabilities are added together to create a new combined probability, \( p_1 + p_3 + p_4 = 0.60 \).

These calculations demonstrate how to handle categorical probabilities in practical scenarios.