At a car rental agency, 0.34 of the cars are returned on time. A sample of 13 car rentals is studied. What is the probability that more than 3 of them are returned on time?
Write only a number as your answer. Round to 2 decimal places ( for example: 0.24). Do not write as a percentage.
To find the probability that more than 3 of the cars are returned on time, we need to calculate the cumulative probability of having 0, 1, 2, and 3 cars returned on time, and then subtract that value from 1.
First, we need to calculate the probability of having exactly x cars returned on time, where x ranges from 0 to 3.
The probability of having 0 cars returned on time can be calculated using the binomial probability formula:
P(X = 0) = (n Choose x) * p^x * (1-p)^(n-x)
where n = 13 (the sample size), p = 0.34 (the probability of a car being returned on time), and x = 0.
Using this formula, the probability of having 0 cars returned on time is:
P(X = 0) = (13 Choose 0) * 0.34^0 * (1-0.34)^(13-0)
Next, we can calculate the probabilities for having 1, 2, and 3 cars returned on time using the same formula:
P(X = 1) = (13 Choose 1) * 0.34^1 * (1-0.34)^(13-1)
P(X = 2) = (13 Choose 2) * 0.34^2 * (1-0.34)^(13-2)
P(X = 3) = (13 Choose 3) * 0.34^3 * (1-0.34)^(13-3)
To calculate the cumulative probability of having at most 3 cars returned on time, we add up these probabilities:
P(X <= 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
Finally, to find the probability of having more than 3 cars returned on time, we subtract the cumulative probability from 1:
P(X > 3) = 1 - P(X <= 3)
Calculating these values, we find:
P(X = 0) = 0.006
P(X = 1) = 0.037
P(X = 2) = 0.097
P(X = 3) = 0.163
P(X <= 3) = 0.006 + 0.037 + 0.097 + 0.163 = 0.303
P(X > 3) = 1 - 0.303 = 0.697
Therefore, the probability that more than 3 of the car rentals are returned on time is approximately 0.70.
To find the probability that more than 3 of the car rentals are returned on time, we need to calculate the probability of the complementary event - the probability that 3 or fewer car rentals are returned on time - and subtract it from 1.
The probability of getting exactly k successes out of n trials in a binomial experiment can be calculated using the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where n is the total number of trials, k is the number of successes, and p is the probability of success in each trial.
In this case, we want to calculate the probability of getting 3 or fewer car rentals returned on time out of 13. The probability of a car being returned on time is 0.34. So the probability of getting exactly k car rentals returned on time can be calculated as:
P(X = k) = (13 choose k) * (0.34)^k * (1 - 0.34)^(13 - k)
To find the probability of getting 3 or fewer car rentals returned on time, we need to sum up the probabilities for k = 0, 1, 2, and 3:
P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
P(X ≤ 3) = [(13 choose 0) * (0.34)^0 * (1 - 0.34)^(13 - 0)] + [(13 choose 1) * (0.34)^1 * (1 - 0.34)^(13 - 1)] + [(13 choose 2) * (0.34)^2 * (1 - 0.34)^(13 - 2)] + [(13 choose 3) * (0.34)^3 * (1 - 0.34)^(13 - 3)]
After calculating the sum of these probabilities, we can subtract the result from 1 to find the probability of more than 3 car rentals being returned on time:
P(X > 3) = 1 - P(X ≤ 3)
Now you can plug in the values and calculate the probability of more than 3 car rentals being returned on time.