Question

Suppose we draw three balls from an urn containing two red balls and three black balls. We do not replace the balls after we draw them. In terms of the hypergeometric distribution, what is the probability of getting two red balls? Compute this probability.

Short answer

0.3

Step-by-step solution

Understand the Hypergeometric Distribution

The hypergeometric distribution describes the probability of k successes (red balls in this case) in n draws (balls drawn) from a finite population without replacement. Here, 'success' refers to drawing a red ball.

Identify the Parameters

Let's identify the parameters for the hypergeometric distribution: - Total population size ( N ): 5 balls (2 red + 3 black) - Total number of success states in the population (red balls, K ): 2 - Total number of draws ( n ): 3 - Number of observed successes (desired number of red balls, k ): 2.

Write the Hypergeometric Probability Formula

The probability of getting exactly 2 red balls out of 3 draws is given by the hypergeometric distribution formula:\[P(X = k) = \frac{{\binom{K}{k} \binom{N-K}{n-k}}}{\binom{N}{n}}\]Where \( \binom{n}{k} \) is the binomial coefficient.

Compute the Binomial Coefficients

Calculate each part of the hypergeometric formula:- \( \binom{2}{2} = 1 \) (ways to choose 2 red balls from 2)- \( \binom{3}{1} = 3 \) (ways to choose 1 black ball from 3)- \( \binom{5}{3} = 10 \) (ways to choose any 3 balls from 5).

Calculate the Probability

Using the hypergeometric probability formula:\[P(X = 2) = \frac{{1 \times 3}}{10} = \frac{3}{10} = 0.3\]

Final Solution

Hence, the probability of drawing exactly 2 red balls from the urn is 0.3.

Key concepts

Probability of Success

In our example, the probability of success refers to the likelihood of drawing a red ball from the urn. Since we are dealing with a hypergeometric distribution, it’s essential to understand that each draw is done without replacement. This means the probability changes with each consecutive draw, unlike the constant probability in the binomial distribution, which involves replacement.
The formula for the probability of success is:

• Find the total number of possible successful outcomes, i.e., drawing the desired number of red balls.
• Determine the total number of outcomes, which is all possible ways to draw a given number of balls from the urn.
• Divide the number of successful outcomes by the total number of outcomes to find the probability.

For the exercise, since we want to find the probability of drawing 2 red balls from a total of 5 balls, with a total of 3 draws, we start by calculating these outcomes, where success specifically means drawing red balls. Simplifying such steps helps understand the practical application of probability in the hypergeometric distribution.

Binomial Coefficient

The binomial coefficient is a crucial part of calculating outcomes in a hypergeometric distribution. It is often represented as \( \binom{n}{k} \), where \( n \) is the total population or available options, and \( k \) is the number needed to be chosen. This coefficient is simply a way to count combinations, not permutations, so the order of selection does not matter.
In our context:

• \( \binom{K}{k} \) represents the ways to choose a specified number of red balls from the total reds available.
• \( \binom{N-K}{n-k} \) represents the ways to choose the remaining balls from the black balls.
• \( \binom{N}{n} \) shows the total ways to select the drawn balls from the whole urn.

Therefore, using the relationship between these binomial coefficients allows for calculating the specific probability of drawing exactly a certain number of successful outcomes (e.g., 2 red balls), embedded into the hypergeometric distribution formula.

Finite Population Sampling

Finite population sampling is central to understanding how the hypergeometric distribution functions. This concept implies sampling from a limited or fixed group without replacement. Each sample affects the composition of the population for subsequent draws.
This kind of sampling contrasts with infinite or large population sampling, where probability distributions assume replacements, like with the binomial distribution. Here, the hypergeometric distribution precisely models scenarios where each choice impacts subsequent probabilities.
In our problem, we sample from a total of 5 balls. Each draw affects the remaining pool of balls, thus changing the probabilities dynamically.

• When a red ball is drawn, the count for red decreases, impacting the likelihood of another red appearing.
• This effect makes finite population sampling ideal for applications requiring real-world sampling and probability predictions, such as our exercise.

Understanding finite population sampling is essential for correctly applying the hypergeometric formula, reflecting real probabilities in small, exact populations.