A. R is a binomial random variable because it satisfies the properties of a binomial experiment:
1. There are a fixed number of trials (5 passengers pressing the button).
2. Each trial has only two outcomes: red light or green light.
3. The probability of getting a red light is constant for each trial (0.30 in this case).
4. The trials are independent of each other.
B. The histogram of the probability distribution of R will have a shape that resembles a binomial distribution. It will be skewed to the right, with the peak around the most probable outcome (in this case, 1 or 2 red lights out of 5 passengers). As the number of red lights increases, the probability decreases, leading to a decreasing tail to the right.
C. To find the probability that exactly 3 out of 5 passengers get a red light, you can use the binomial probability formula:
P(R = 3) = (5 choose 3) * (0.30)^3 * (0.70)^(5-3)
P(R = 3) = 10 * (0.30)^3 * (0.70)^2
P(R = 3) = 10 * 0.027 * 0.49
P(R = 3) = 0.1323
Therefore, the probability that exactly 3 out of 5 passengers get a red light is approximately 0.1323 or 13.23%.
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