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Chapter 4: Problem 32

Express all probabilities as fractions. Mendel conducted some 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.

Short Answer

Expert verified
The probability that the four selected peas are of the same type is \( \frac{1}{35} \).

Step by step solution

01

- Understand the Total Sample Space

First, determine the total number of ways to select 4 peas from the 8 available peas. This can be calculated using the combination formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]Here, \(n = 8\) and \(k = 4\). Calculate: \[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]So, there are 70 possible ways to choose 4 peas out of 8.
02

- Calculate the Favorable Outcomes for Smooth Yellow Peas

Next, find the number of ways to select 4 smooth yellow peas from the 4 available smooth yellow peas. This is also a combination and can be calculated as: \[ \binom{4}{4} = \frac{4!}{4!(4-4)!} = \frac{4!}{4!0!} = 1 \]There is only 1 way to choose 4 smooth yellow peas from 4.
03

- Calculate the Favorable Outcomes for Wrinkly Green Peas

Now, find the number of ways to select 4 wrinkly green peas from the 4 available wrinkly green peas. Similar to the previous step, calculate: \[ \binom{4}{4} = \frac{4!}{4!(4-4)!} = \frac{4!}{4!0!} = 1 \]There is also only 1 way to choose 4 wrinkly green peas from 4.
04

- Calculate the Total Favorable Outcomes

Add the favorable outcomes from both smooth yellow peas and wrinkly green peas: \[ 1 + 1 = 2 \]So, there are 2 favorable ways out of the total 70 ways to choose 4 peas that are of the same type.
05

- Calculate the Probability

Finally, calculate the probability that the 4 selected peas are of the same type. Use the fraction of favorable outcomes over the total outcomes: \[ \frac{2}{70} = \frac{1}{35} \]Thus, the probability is \( \frac{1}{35} \).

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Key Concepts

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

Combination Formula
The combination formula is a fundamental concept in probability and statistics. It helps us determine how many ways we can choose a subset of items from a larger set without regard to the order of selection. The formula is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

Here:
  • \(n\) represents the total number of items.
  • \(k\) represents the number of items to choose.
  • \(!\) denotes factorial, which is the product of all positive integers up to that number. For example, \(!4 = 4 脳 3 脳 2 脳 1 = 24\).

In the example given, we have 8 peas (\(n = 8\)) and want to choose 4 of them (\(k = 4\)). Using the formula, we calculate:
\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{40320}{24 \times 24} = 70 \]
This shows there are 70 different ways to select 4 peas from a total of 8.
Sample Space
The sample space in probability refers to the set of all possible outcomes of an experiment. It's essential for determining probabilities because it gives us the context for calculating favorable outcomes.

For the problem at hand, the sample space is the total number of ways to select 4 peas from a batch of 8 (4 smooth yellow and 4 wrinkly green). We previously calculated that there are 70 possible ways to do this.

Understanding the sample space helps us see all possible scenarios. It enables us to identify and count the number of favorable outcomes among these possibilities.
Favorable Outcomes
Favorable outcomes are the specific outcomes that satisfy the condition of our probability question. In this problem, we want to know the probability that all 4 selected peas are of the same type.

We have two types of peas: smooth yellow and wrinkly green. There is only one way to choose 4 smooth yellow peas from the 4 available, calculated as follows:
\[ \binom{4}{4} = \frac{4!}{4!(4-4)!} = \frac{24}{24 \times 1} = 1 \]
Similarly, there is only one way to choose 4 wrinkly green peas from the 4 available:
\[ \binom{4}{4} = \frac{4!}{4!(4-4)!} = \frac{24}{24 \times 1} = 1 \]
Therefore, the total number of favorable outcomes is 1 (smooth yellow) + 1 (wrinkly green) = 2.
Elementary Statistics
Elementary statistics involves the basic concepts and techniques used to summarize and make inferences about data. A key component is understanding probabilities, which is the likelihood of an event occurring.

In this exercise, we used elementary statistics methods to calculate the probability of a specific event: selecting 4 peas of the same type from a batch. Here's a step-by-step summary:
  • First, calculate the total number of possible outcomes using the combination formula.
  • Next, determine the number of favorable outcomes (same type peas).
  • Finally, find the probability by dividing the number of favorable outcomes by the total number of possible outcomes.

Thus, we calculated that the probability is \(\frac{1}{35}\). Understanding these basic steps and concepts helps build a strong foundation in elementary statistics and probability.

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Most popular questions from this chapter

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