One of Mendel’s famous experiments with peas resulted in 580 offspring, and 152 of them were yellow peas. Mendel claimed that under the same conditions, 25% of offspring peas would be yellow. Assume that Mendel’s claim of 25% is true, and assume that a sample consists of 580 offspring peas.

b. Find the probability of exactly 152 yellow peas.

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Step 1: Identify the type of probability distribution to use. Since we are dealing with a fixed number of trials (580 peas), each trial has two possible outcomes (yellow or not yellow), and the probability of success (yellow pea) is constant (25%), this is a binomial probability problem.

Step 2: Write the formula for the binomial probability distribution. The probability of exactly k successes in n trials is given by:

P(X = k) = C(n, k) * p

^k * (1 - p)^n-k, where C(n, k) is the binomial coefficient, p is the probability of success, and n is the number of trials.

Step 3: Substitute the given values into the formula. Here, n = 580 (total peas), k = 152 (yellow peas), and p = 0.25 (probability of a yellow pea). The binomial coefficient C(n, k) is calculated as:

C(n, k) = n! / (k! * (n - k)!)

Step 4: Compute the binomial coefficient C(580, 152). This involves factorial calculations for 580, 152, and (580 - 152). Then, multiply the result by

p

^k and

(1 - p)

^n-k.

Step 5: Use a calculator or statistical software to evaluate the final probability. Since factorials for large numbers can be computationally intensive, it is often practical to use software or a binomial probability table to find the result.

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주요 개념

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Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the 'success' is the occurrence of yellow peas, with a probability of 25%. The distribution is defined by two parameters: the number of trials (n) and the probability of success (p).

Probability Mass Function (PMF)

The probability mass function gives the probability of obtaining exactly k successes in n trials for a binomial distribution. It is calculated using the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k). This function is essential for determining the likelihood of observing exactly 152 yellow peas out of 580.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this scenario, we can test Mendel's claim that 25% of the offspring are yellow peas by comparing the observed number of yellow peas (152) to the expected number based on the binomial distribution. This helps assess whether the observed result is consistent with the hypothesis.