Let's analyze and solve problem 5(b) based on the content extracted from the image.

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Problem 5(b) Description:

The problem seems to be asking:

Given that a player has 5 coins, each of which can show either heads (H) or tails (T), what is the probability that exactly 3 of these coins show heads?

Solution:

1. Determine the total number of possible outcomes:

   • Since each coin can show either heads or tails, and there are 5 coins, the total number of possible outcomes is: $2^5 = 32$

2. Calculate the number of favorable outcomes:

   • We need exactly 3 heads. The number of ways to choose 3 heads out of 5 coins is given by the combination formula: $\text{Number of favorable outcomes} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10$

3. Calculate the probability:

   • The probability of getting exactly 3 heads is the ratio of the number of favorable outcomes to the total number of outcomes: $\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{10}{32} = \frac{5}{16}$

Final Answer:

The probability that exactly 3 out of the 5 coins show heads is $\frac{5}{16}$.

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Would you like more details or have any questions on this problem? Here are some related questions that you might find helpful:

1. How do you calculate combinations like $\binom{5}{3}$?
2. What is the formula for calculating the total number of outcomes in a probability scenario?
3. How does the concept of independent events apply to this problem?
4. What would be the probability if we needed exactly 2 heads instead of 3?
5. Can you explain the binomial distribution and how it relates to this problem?
6. What would happen to the probability if there were 6 coins instead of 5?
7. How can we verify the answer using a different method, such as listing all possible outcomes?
8. What if we wanted to find the probability of getting at least 3 heads?

Tip: When dealing with probabilities, always ensure that the total number of outcomes is correctly calculated, as any mistake there can lead to incorrect results.