Question

A commuter encounters four traffic lights each day on her way to work. Let X represent the number of these that are red lights. The probability mass function of X is as follows.x 0 1 2 3 4P(X = x) 0.1 0.2 0.4 0.2 0.1what is the probability that in a period of 100 days, the average number of red lights encountered is more than 2 per day?

Answer 1

The probability that the average number of red lights encountered in 100 days is more than 2 per day is approximately 0.00001894, or 0.001894%.

Step-by-step explanation:

To find the probability that the average number of red lights encountered in 100 days is more than 2 per day, we can follow these steps:

Calculate the average number of red lights per day: Multiply each possible number of red lights by its corresponding probability and sum the results.

Determine the desired probability: Find the probability of the average number of red lights exceeding 2 per day.

Here's the detailed calculation:

1. Average number of red lights per day:

Average = 0 × 0.1 + 1 × 0.2 + 2 × 0.4 + 3 × 0.2 + 4 × 0.1 = 1.8

2. Desired probability:

Since the average number of red lights per day is 1.8, we need to find the probability of encountering more than 2 red lights on average in 100 days. This can be achieved by summing the probabilities of encountering 3 or 4 red lights on any given day and raising the sum to the power of 100 (due to the 100-day period):

P(average > 2) = (0.2 + 0.1)¹⁰⁰ ≈ 0.00001894

Therefore, the probability that the average number of red lights encountered in 100 days is more than 2 per day is approximately 0.00001894, or 0.001894%.

Answer 2

To find the probability that in a period of 100 days, the average number of red lights encountered is more than 2 per day, we need to sum the probabilities for 3 and 4 and subtract the probability for 2.

Step-by-step explanation:

To find the probability that in a period of 100 days, the average number of red lights encountered is more than 2 per day, we need to find the probability that the average number of red lights is greater than 2.

The average number of red lights encountered in 100 days would be 100 times the average number of red lights encountered per day.

The average number of red lights encountered per day can range from 0 to 4, according to the probability mass function given. So, we need to find the probability that the average number of red lights is greater than 2 by summing the probabilities for 3 and 4 and subtracting the probability for 2.

The probability that the average number of red lights encountered per day is greater than 2 is P(X = 3) + P(X = 4) = 0.2 + 0.1 = 0.3. Therefore, the probability that in a period of 100 days, the average number of red lights encountered is more than 2 per day is 0.3.