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Chapter 4: Problem 27

Find the probability and answer the questionsWhen Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 428 green peas and 152 yellow peas. Based on those results, estimate the probability of getting an offspring pea that is green. Is the result reasonably close to the expected value of \(3 / 4,\) as Mendel claimed?

Short Answer

Expert verified
The probability of a green pea is approximately 0.7379, which is reasonably close to Mendel's expected probability of 0.75.

Step by step solution

01

- Determine Total Sample Size

Add the number of green peas and yellow peas to find the total sample size. The total sample size is the sum of green peas (428) and yellow peas (152). So, 428 + 152 = 580.
02

- Calculate the Probability of Green Peas

To find the probability of a pea being green, divide the number of green peas by the total sample size. Using the numbers, the probability is \( \frac{428}{580} \).
03

- Simplify the Probability

Simplify the fraction \( \frac{428}{580} \) by dividing both the numerator and the denominator by their greatest common divisor (which is 4). Thus, \( \frac{428 \div 4}{580 \div 4} = \frac{107}{145} \).
04

- Convert Fraction to Decimal

Convert the fraction \( \frac{107}{145} \) to a decimal by performing the division. \( \frac{107}{145} \approx 0.7379 \). This means the probability of getting a green pea is approximately 0.7379.
05

- Compare with Expected Probability

The expected probability according to Mendel is \( \frac{3}{4} \) or 0.75. Compare 0.7379 with 0.75. Since they are very close, we can say that the observed probability is reasonably close to the expected probability.

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Key Concepts

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

Mendel's genetics
Mendel's genetics is the study of how traits are inherited from one generation to the next. Gregor Mendel, through his work on pea plants, discovered the fundamental laws of inheritance. He found that traits are determined by pairs of genes, one from each parent, and these traits can be either dominant or recessive. This foundational work is critical for understanding genetic probability in biology.
probability calculation
Probability calculation helps us determine how likely an event is to occur. In Mendel's experiments, we calculate the probability of getting a green pea by dividing the number of green peas by the total number of peas. Probability ranges from 0 to 1, where 0 means the event will not happen, and 1 means it will definitely happen. For example, the probability of selecting a green pea from 428 green and 152 yellow peas is calculated as follows: \(\frac{428}{580}\).
sample size analysis
Sample size analysis is crucial in statistics because a larger sample size can give a more accurate estimate of the true probability. In Mendel's experiment, our sample consists of 580 peas, combining 428 green and 152 yellow peas. Larger samples tend to yield results that are closer to what is expected based on theoretical probabilities, minimizing the effect of random variations.
fraction simplification
Fraction simplification makes probabilities easier to understand and compare. In our example, the probability of getting a green pea is initially \( \frac{428}{580} \). To simplify, we find the greatest common divisor (GCD) of 428 and 580, which is 4. Dividing both the numerator and the denominator by 4 gives us \( \frac{107}{145} \).
decimal conversion
Decimal conversion translates a fraction into its decimal form, making it easier to interpret and compare. To convert \( \frac{107}{145} \) into a decimal, we perform the division \( 107 \div 145 \approx 0.7379 \). This indicates that the probability of getting a green pea is approximately 0.7379, which we can compare to the expected probability of 0.75. The close values show that our results align well with Mendel's expected probability.

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Most popular questions from this chapter

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