The question in the image asks about the mistake in using the multiplication rule for finding the probability of selecting two peas with green pods followed by three peas with yellow pods. The probability calculation presented is $(0.62)(0.62)(0.38)(0.38)(0.38) = 0.02109$, and we are supposed to determine the flaw in this reasoning.

Key Points:

• 62% of peas have green pods, and 38% have yellow pods.
• The multiplication rule assumes a specific sequence: first two peas being green, followed by three peas being yellow.
• The error lies in using this approach for finding the probability of exactly two green peas and three yellow peas, as there are multiple ways (permutations) of achieving this combination.

Correct Answer:

The correct answer is A:

• "The probability obtained in this way is too high, since it only accounts for the permutation of getting two green followed by three yellow. There are many other permutations through which totals of two green and three yellow can be obtained."

Explanation:

• The calculation only considers one specific sequence: two green peas followed by three yellow peas.
• However, the problem asks for the probability of getting exactly two green peas and three yellow peas in any order, which involves calculating the number of possible permutations of the green and yellow peas and adjusting the probability accordingly.

Would you like more details or have any questions?

Here are some follow-up questions to deepen your understanding:

1. What is the difference between permutations and combinations in probability?
2. How would you calculate the number of possible ways to arrange two green peas and three yellow peas in any order?
3. How would the probability change if the peas were sampled without replacement?
4. Why does the multiplication rule apply only to specific sequences rather than unordered selections?
5. What additional steps are necessary to account for multiple permutations in probability problems?

Tip: When solving probability problems involving specific combinations of outcomes, always account for all possible permutations unless the problem specifies a fixed order.