y	P(Y=y)
0	0.187
1	0.274
2	0.219
3	0.094
4	0.042
5	0.043
6	0.021
7	0.013
8	0.107 d. Use the special addition rule and the probability distribution to determine the probability that the year had at least one major hurricane.
The probability is 
  
enter your response here. 
​(Type an integer or decimal rounded to three decimal places as​ needed.)
Part 5
e. Use the special addition rule and the probability distribution to determine the probability that the year had exactly two major hurricanes.
The probability is 
  
enter your response here. 
​(Type an integer or decimal rounded to three decimal places as​ needed.)
Part 6
f. Use the special addition rule and the probability distribution to determine the probability that the year had between 3 and 5 major​ hurricanes, inclusive.
The probability is 
  
enter your response here.
​(Type an integer or decimal rounded to three decimal places as​ needed.)
Complete parts (a) through (f) below.Complete parts (a) through (f) below.Select the correct choice below and fill in the answer box within your choice.input field 3input field 4Select the correct choice below and fill in the answer box within your choice.input field 5input field 6Select the correct choice below and fill in the answer box within your choice.The probability is(Type an integer or decimal rounded to three decimal places as needed.)The probability is(Type an integer or decimal rounded to three decimal places as needed.)The probability is(Type an integer or decimal rounded to three decimal places as needed.)


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:
\[
\boxed{0.179}
\]

---

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!

Use the special addition rule and the probability distribution to determine the probability that the year had at least one major hurricane, exactly two major hurricanes, and between 3 and 5 major hurricanes, inclusive. The probabilities provided are: P(Y=0)=0.187, P(Y=1)=0.274, P(Y=2)=0.219, P(Y=3)=0.094, P(Y=4)=0.042, P(Y=5)=0.043, P(Y=6)=0.021, P(Y=7)=0.013, P(Y=8)=0.107.

Explore how to use the special addition rule and the complement rule in probability to find the likelihood of major hurricanes in a given year. This example covers at least one hurricane, exactly two hurricanes, and between 3 and 5 hurricanes using a given probability distribution.

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