6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
4:57 minutesProblem 6.6.17a
Textbook Question
Mendelian Genetics When Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 929 peas, with 705 of them having red flowers. If we assume, as Mendel did, that under these circumstances, there is a 3/4 probability that a pea will have a red flower, we would expect that 696.75 (or about 697) of the peas would have red flowers, so the result of 705 peas with red flowers is more than expected.
a. If Mendel’s assumed probability is correct, find the probability of getting 705 or more peas with red flowers.
Verified step by step guidance1
Step 1: Identify the type of probability distribution. Since the problem involves a fixed number of trials (929 peas), two possible outcomes (red flower or not), and a constant probability of success (3/4 for red flowers), this is a binomial distribution problem.
Step 2: Define the parameters of the binomial distribution. The number of trials (n) is 929, the probability of success (p) is 3/4, and the number of successes (x) is 705 or more.
Step 3: Use the complement rule to simplify the calculation. The probability of getting 705 or more peas with red flowers is equivalent to 1 minus the probability of getting fewer than 705 peas with red flowers. Mathematically, this can be expressed as P(X ≥ 705) = 1 - P(X < 705).
Step 4: Convert the cumulative probability into a summation. For a binomial distribution, P(X < 705) is the sum of probabilities for all values of X from 0 to 704. This can be written as P(X < 705) = Σ P(X = k) for k = 0 to 704, where P(X = k) = (n choose k) * p^k * (1-p)^(n-k).
Step 5: Use a normal approximation to the binomial distribution for computational simplicity. Since n is large, the binomial distribution can be approximated by a normal distribution with mean μ = n * p and standard deviation σ = √(n * p * (1-p)). Apply the continuity correction by finding P(X ≥ 704.5) using the normal distribution formula Z = (X - μ) / σ, and calculate the corresponding probability from the standard normal table.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability Distribution
A probability distribution describes how the probabilities are distributed over the values of a random variable. In this context, Mendel's assumption of a 3/4 probability for red flowers suggests a binomial distribution, where each pea can be seen as a trial with two outcomes: red or not red. Understanding this distribution is crucial for calculating the likelihood of observing a specific number of successes, such as 705 red flowers.
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Binomial Probability Formula
The binomial probability formula calculates the probability of obtaining a fixed number of successes in a given number of independent Bernoulli trials. It is expressed as P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n' is the total number of trials, 'k' is the number of successes, and 'p' is the probability of success. This formula is essential for determining the probability of getting 705 or more red flowers in Mendel's experiment.
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Normal Approximation to the Binomial
For large sample sizes, the binomial distribution can be approximated by a normal distribution, which simplifies calculations. This approximation is valid when both np and n(1-p) are greater than 5. In Mendel's case, with 929 peas, this approximation allows for easier computation of probabilities, particularly when determining the likelihood of observing 705 or more red flowers.
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