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Ch. 4 - Probability
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 4 - ProbabilityProblem 4.4.32
Chapter 4, Problem 4.4.32

Mendel’s Peas Mendel conducted some of his famous experiments with peas that were either smooth yellow plants or wrinkly green plants. If four peas are randomly selected from a batch consisting of four smooth yellow plants and four wrinkly green plants, find the probability that the four selected peas are of the same type.

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Step 1: Understand the problem. We are tasked with finding the probability that all four selected peas are of the same type (either all smooth yellow or all wrinkly green) when randomly selecting four peas from a batch of 4 smooth yellow and 4 wrinkly green peas. This is a problem involving combinations and probabilities.
Step 2: Calculate the total number of ways to select 4 peas from the batch. The total number of peas is 8 (4 smooth yellow + 4 wrinkly green). The total number of combinations to select 4 peas from 8 is given by the combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items and \( r \) is the number of items to choose. Here, \( n = 8 \) and \( r = 4 \).
Step 3: Calculate the number of favorable outcomes for selecting 4 peas of the same type. There are two cases: (1) all 4 peas are smooth yellow, or (2) all 4 peas are wrinkly green. For each case, the number of ways to select 4 peas of the same type is \( \binom{4}{4} = 1 \), since there are only 4 peas of each type available.
Step 4: Add the favorable outcomes from both cases. Since the two cases are mutually exclusive, the total number of favorable outcomes is \( 1 + 1 = 2 \).
Step 5: Calculate the probability. The probability is the ratio of favorable outcomes to total outcomes. Using the results from Steps 2 and 4, the probability is \( P = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{2}{\binom{8}{4}} \). Simplify this expression to find the final probability.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it involves calculating the chances of selecting four peas of the same type from a mixed batch. Understanding basic probability principles, such as favorable outcomes versus total outcomes, is essential for solving the problem.
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Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. In this scenario, it helps determine how many ways we can select the peas from the two types available. This concept is crucial for calculating the total number of possible selections and the specific selections that meet the criteria of the problem.

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this case, it can be applied to model the selection of peas, where each pea can be classified as either smooth yellow or wrinkly green. Understanding this distribution aids in calculating the probabilities of selecting all peas of one type.
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Related Practice
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