91Ó°ÊÓ

Gregor Mendel, often called the father of genetics, performed groundbreaking experiments with pea plants to understand how traits are inherited. In one of his famous experiments, he dealt with flower color, specifically red and white flowers. He theorized that each pea plant had a 3/4 probability of producing red flowers. This probability was derived from Mendel's law of segregation, which suggests that traits are passed down from parents to offspring in specific ratios.
To test his theories, Mendel conducted multiple trials and collected large samples of peas to observe the actual outcomes. For example, in one experiment with 929 offspring, 705 peas had red flowers, slightly above the expected number based on the 3/4 probability.
His approach in counting and classifying these traits laid the groundwork for modern genetics, enabling scientists to predict inheritance patterns using probabilities.

The binomial distribution represents the number of successes in a fixed number of independent trials, each with the same probability of success. In Mendel's experiment, each pea plant represents a trial, and having a red flower is considered a success.
However, calculating probabilities directly from the binomial distribution can be complex, especially with large sample sizes. Instead, we can use the **normal approximation** to the binomial distribution, leveraging the Central Limit Theorem. This theorem states that the binomial distribution approaches a normal distribution as the number of trials becomes large.
To apply the normal approximation, we need to compute the mean \(\big(\text{\mu}\big)\) and the standard deviation \(\big(\sigma\big)\) of the binomial distribution. Then, we can transform our binomial distribution into a standard normal distribution, simplifying the probability calculations.

A z-score allows us to determine how many standard deviations an observation (in this case, the number of pea plants with red flowers) is from the mean of the distribution.
To calculate the **z-score**, we use the formula:

\[ z = \frac{x - \mu}{\sigma} \]
where \(x\big)\) is the observed value, \(\big(\mu\big)\) is the mean, and \(\big(\sigma\big)\) is the standard deviation.
For Mendel's experiment:
\[ \mu = 929 \cdot \frac{3}{4} \approx 696.75 \]
\[ \sigma = \sqrt{929 \cdot \frac{3}{4} \cdot \big(\frac{1}{4}\big)} \approx 13.27 \]
Given Mendel observed 705 pea plants with red flowers, the z-score is calculated as:
\[ z = \frac{705 - 696.75}{13.27} \approx 0.624 \]
This z-score tells us how far away 705 is from the expected mean in standard deviation terms.

To understand whether the observed number of red-flowered pea plants is significantly different from the expected number, we assess the probability of obtaining such a result. After calculating the z-score, we can use the standard normal distribution table to find the corresponding probability.
For our calculated z-score of approximately 0.624, the area to the left of this z-score (which represents the cumulative probability) is roughly 0.733. Since we are interested in the probability of having 705 or more red-flowered peas, we find the complement:

\[ P(X \ge 705) = 1 - 0.733 = 0.267 \]
This result means there is a 26.7% chance of observing at least 705 pea plants with red flowers.
To determine if this is significantly high, we compare it to a common threshold, often set at 0.05 (or 5%). Since 0.267 is much greater than 0.05, 705 red-flowered peas is not considered significantly high, suggesting Mendel's assumption of a \(\frac{3}{4}\big)\) probability for red flowers is plausible.