Binomial experiment

Explanation This is a binomial experiment because it meets all of the following criteria:

• The experiment has a fixed number of trials, n = 20.
• Each trial has only two outcomes, red or not red.
• The probability of success (red) remains constant from trial to trial, p = 0.22.
• The trials are independent, meaning that the outcome of one trial does not affect the outcome of another trial.

Probability that exactly 6 cars are red The probability that exactly 6 cars are red is given by the binomial probability formula: P(X=6) = nCr * p^r * (1-p)^(n-r) where

• n = 20 is the number of trials
• r = 6 is the number of successes (red cars)
• p = 0.22 is the probability of success (red car)
• (1-p) = 0.78 is the probability of failure (not red car)

Plugging these values into the formula, we get: P(X=6) = 20C6 * (0.22)^6 * (0.78)^14 ≈ 0.0277 Interpretation The probability that exactly 6 cars are red is 0.0277, or about 2.77%. This means that in 100 trials, we would expect to see exactly 6 cars that are red about 3 times. Probability that fewer than 6 cars are red The probability that fewer than 6 cars are red is given by: P(X<6) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) where

• P(X=0) is the probability that no cars are red
• P(X=1) is the probability that 1 car is red
• etc.

We can calculate these probabilities using the binomial probability formula. The results are shown below: P(X=0) = 20C0 * (0.22)^0 * (0.78)^20 = 0.0547 P(X=1) = 20C1 * (0.22)^1 * (0.78)^19 = 0.1968 P(X=2) = 20C2 * (0.22)^2 * (0.78)^18 = 0.4613 P(X=3) = 20C3 * (0.22)^3 * (0.78)^17 = 0.6678 P(X=4) = 20C4 * (0.22)^4 * (0.78)^16 = 0.6366 P(X=5) = 20C5 * (0.22)^5 * (0.78)^15 = 0.3942 Adding these probabilities, we get: P(X<6) = 0.0547 + 0.1968 + 0.4613 + 0.6678 + 0.6366 + 0.3942 = 1.8564 Interpretation The probability that fewer than 6 cars are red is 1.8564, or about 18.56%. This means that in 100 trials, we would expect to see fewer than 6 cars that are red about 18 times. Probability that at least 6 cars are red The probability that at least 6 cars are red is given by: 1 - P(X<6) = 1 - 1.8564 = 0.1436 Interpretation The probability that at least 6 cars are red is 0.1436, or about 14.36%. This means that in 100 trials, we would expect to see at least 6 cars that are red about 14 times. Mean and standard deviation T he mean and standard deviation of the binomial random variable are given by: Mean = np = 20 * 0.22 = 4.4Standard deviation = np(1-p) = 4.4 * 0.78 = 3.40 Interpretation The mean of the binomial random variable is 4.4, which means that we would expect to see an average of 4.4 red cars in 20 trials. The standard deviation of the binomial random variable is 3.40, which means that there is a 68% chance that the number of red cars will be within 1 standard deviation of the mean, or between 1 and 7.4 red cars.

Sample Solution

This is a binomial experiment because it meets all of the following criteria: