Let's go through each part systematically.

Part d: Probability that the year had at least one major hurricane

To find the probability that there was at least one major hurricane, we need to use the complement rule. The complement rule states that the probability of at least one event occurring is $P(\text{at least one hurricane}) = 1 - P(\text{no hurricanes})$.

From the table, the probability that the year had no hurricanes is given as $P(Y = 0) = 0.187$.

Thus, the probability of at least one major hurricane is: $P(\text{at least one hurricane}) = 1 - 0.187 = 0.813$

So, the probability is: $\boxed{0.813}$

Part e: Probability that the year had exactly two major hurricanes

From the table, the probability that the year had exactly two major hurricanes is: $P(Y = 2) = 0.219$

So, the probability is: $\boxed{0.219}$

Part f: Probability that the year had between 3 and 5 major hurricanes (inclusive)

To find the probability that the year had between 3 and 5 major hurricanes, inclusive, we need to sum the probabilities for $Y = 3$, $Y = 4$, and $Y = 5$.

From the table:

• $P(Y = 3) = 0.094$
• $P(Y = 4) = 0.042$
• $P(Y = 5) = 0.043$

Thus, the total probability is: $P(3 \leq Y \leq 5) = 0.094 + 0.042 + 0.043 = 0.179$

So, the probability is: $\boxed{0.179}$

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Let me know if you would like further details or explanations!

Here are 5 related questions:

1. What is the total probability for the given distribution?
2. What is the probability of having more than 5 major hurricanes in a year?
3. What is the probability of having fewer than 2 major hurricanes in a year?
4. How would the complement rule apply if we wanted to calculate "no more than 2 hurricanes"?
5. What is the expected number of major hurricanes in a given year?

Tip: For probabilities involving ranges, simply add the individual probabilities within the specified range!