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Let's analyze the problem from the image you uploaded, which involves a probability distribution for the number of speeding tickets a person receives in a three-year period.

### Table Information
You are given the following probability distribution table, where \( P(x) \) represents the probability of receiving exactly \( x \) tickets:

| \( x \) (Number of tickets) | \( P(x) \) (Probability) |
|-----------------------------|--------------------------|
| 0                           | 0.341                    |
| 1                           | 0.3065                   |
| 2                           | 0.1831                   |
| 3                           | 0.1157                   |
| 4                           | 0.046                    |
| 5                           | 0.0077                   |
| 6+                          | 0.0000                   |

#### Questions:

1. **(a) What is the probability that a randomly selected person has received four tickets in a three-year period?**
   - This is directly from the table. 
   - \( P(x = 4) = 0.046 \).

2. **(b) What is the probability that a randomly selected person has received four or more tickets in a three-year period?**
   - We need to sum the probabilities for \( x = 4 \) and \( x = 5 \), as \( P(x \geq 6) = 0 \).
   - \( P(x \geq 4) = P(x = 4) + P(x = 5) = 0.046 + 0.0077 = 0.0537 \).

3. **(c) What is the probability that a randomly selected person has received more than four tickets in a three-year period?**
   - This involves only \( x = 5 \), as \( P(x = 6+) = 0 \).
   - \( P(x > 4) = P(x = 5) = 0.0077 \).

4. **(d) Which probability should we use to determine whether it is unusual to randomly select a person who has received four or more tickets in a three-year period?**
   - The correct answer is \( P(x \geq 4) \), since we are interested in the probability of receiving *four or more* tickets.

Let me know if you'd like further details or calculations!

### 5 Related Questions:
1. How is the cumulative probability calculated in probability distribution problems?
2. What is the expected number of tickets a person receives based on this distribution?
3. How would you interpret the probability for receiving zero tickets (P(x=0))?
4. What is the probability of receiving between 1 and 3 tickets, inclusive?
5. How would adding additional values for \( x \geq 6 \) affect the calculations?

### Tip:
Always make sure the total probability in a probability distribution sums up to 1 for the entire sample space.

What is the probability that a randomly selected person has received four tickets in a three-year period? What is the probability that a person has received four or more tickets? What is the probability that a person has received more than four tickets? Which probability should we use to determine whether it is unusual to receive four or more tickets?

Analyze a probability distribution table for speeding tickets received over a three-year period. This problem explores calculating probabilities for exact values and cumulative probabilities, including the probability of receiving four or more speeding tickets. Suitable for Grades 10-12.

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