According to a survey by paint manufacturer, DuPont, 22% of all cars in the United States are red. Suppose 20 cars are randomly selected and the number of red cars are recorded. Round probabilities to 4 decimal places.
Explain why this is a binomial experiment.
Find and interpret the probability that exactly 6 cars are red.
Find and interpret the probability that fewer than 6 cars are red.
Find and interpret the probability that at least 6 cars are red.
Compute the mean and standard deviation of the binomial random variable.
Full Answer Section
- The trials are independent. The outcome of one trial does not affect the outcome of another trial.
- The number of trials is fixed. We are sampling 20 cars.
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^x * (1 - p)^n - x
where:
- n = number of trials = 20
- x = number of successes = 6
- p = probability of success = 0.22
- (1 - p) = probability of failure = 0.78
Plugging these values into the formula, we get:
P(x = 6) = 20C6 * (0.22)^6 * (0.78)^14
= 0.028849
Interpretation
The probability that exactly 6 cars are red is 0.028849, or about 2.9%. This means that if we randomly select 20 cars, we would expect to see exactly 6 cars that are red about 2.9% of the time.
Probability that fewer than 6 cars are red
The probability that fewer than 6 cars are red is given by the following sum:
P(x < 6) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5)
The probabilities for each of these values can be found using the binomial probability formula. The sum of these probabilities is:
P(x < 6) = 0.14333 + 0.30960 + 0.28191 + 0.16999 + 0.07692 + 0.02884
= 0.91259
Interpretation
The probability that fewer than 6 cars are red is 0.91259, or about 91.3%. This means that if we randomly select 20 cars, we would expect to see fewer than 6 cars that are red about 91.3% of the time.
Probability that at least 6 cars are red
The probability that at least 6 cars are red is given by 1 minus the probability that fewer than 6 cars are red. This is:
P(x >= 6) = 1 - 0.91259
= 0.08741
Interpretation
The probability that at least 6 cars are red is 0.08741, or about 8.7%. This means that if we randomly select 20 cars, we would expect to see at least 6 cars that are red about 8.7% of the time.
Mean and standard deviation
The mean of the binomial random variable is given by np. In this case, the mean is 0.22 * 20 = 4.4.
The standard deviation of the binomial random variable is given by np(1 - p). In this case, the standard deviation is 0.22 * 0.78 * 20 = 2.86.
Conclusion
This problem illustrates how to use the binomial probability formula to calculate the probability of specific events in a binomial experiment. It also shows how to calculate the mean and standard deviation of a binomial random variable.