Based on the information provided, it seems that there was a mistake in the calculations for parts (a) and (b) of the question. Let's go through the correct calculation steps together:
(a) To calculate the probability that exactly five students are left-handed, we use the binomial probability formula. The formula is:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of exactly k successes
- n is the total number of trials (31 in this case)
- k is the number of successes (5 in this case)
- p is the probability of success (13%, or 0.13)
Using this formula, we can calculate the probability as follows:
P(X = 5) = (31 choose 5) * 0.13^5 * (1-0.13)^(31-5)
To calculate this, you can use a calculator or a statistical software. If you don't have access to these tools, you can also use a search engine like Google by entering the following expression in the search bar:
(31 choose 5) * 0.13^5 * (1-0.13)^(31-5)
(b) To calculate the probability that at most four students are left-handed, we need to sum the probabilities of having 0, 1, 2, 3, or 4 left-handed students. You can do this by using the same binomial probability formula and calculating each term individually. Then, you add up the results.
(c) To calculate the probability that at least six students are left-handed, you can subtract the probability of having five or fewer left-handed students from 1. So, the calculation would be:
P(X >= 6) = 1 - P(X <= 5)
By following these steps correctly, you should be able to get the correct answers for parts (a), (b), and (c) of the question.