Final Answer:
1. The probability that Mr. Caywood will pick all 32 first-round games correctly is approximately 0.00003 or 0.003%.
2. The probability that Mr. Caywood will pick exactly 3 games correctly in the first round is calculated as 0.2317 or 23.17%.
3. The probability that Mr. Caywood will pick exactly 15 games incorrectly in the first round is approximately 0.0296 or 2.96%.
Step-by-step explanation:
To calculate the probability of Mr. Caywood picking all 32 first-round games correctly, we use the probability of guessing one game correctly (0.54) raised to the power of the number of games (32). This results in an extremely low probability, indicating that successfully predicting all 32 games is a highly unlikely event.
For the probability of picking exactly 3 games correctly, we use the binomial probability formula. This involves calculating the combination of 32 games taken 3 at a time, multiplied by the probability of success (0.54) raised to the power of the number of successes (3), multiplied by the probability of failure (1 - 0.54) raised to the power of the number of failures (32 - 3).
Similarly, for the probability of picking exactly 15 games incorrectly, we use the binomial probability formula. This time, we calculate the combination of 32 games taken 15 at a time, multiplied by the probability of failure (1 - 0.54) raised to the power of the number of failures (15), multiplied by the probability of success (0.54) raised to the power of the number of successes (32 - 15).
In both cases, the binomial probability formula helps us model the likelihood of achieving a specific number of successes or failures in a series of independent events, providing the probabilities for the given scenarios.