The Probability that a randomly selected commuter listens to audiobooks during their commute is 0.49. Suppose that a sample of 10 commuters is selected, what is the probability that at most 3 of them listen to audiobooks?
Write only a number as your answer. Round to 2 decimal places (for example 0.24). Do not write as a percentage.
no i just need help knowing to go about it!
To calculate the probability that at most 3 of the selected commuters listen to audiobooks, we can use the binomial probability formula.
The binomial probability formula is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes.
- n is the number of trials (sample size).
- p is the probability of success on each trial.
- C(n, k) is the binomial coefficient (also known as the combination of n and k).
In this case, we want to find the probability that at most 3 of the 10 commuters listen to audiobooks. This means we need to calculate the probabilities for k = 0, 1, 2, and 3, and sum them up.
Let's calculate each probability separately:
For k = 0:
P(X = 0) = C(10, 0) * 0.49^0 * (1 - 0.49)^(10 - 0)
For k = 1:
P(X = 1) = C(10, 1) * 0.49^1 * (1 - 0.49)^(10 - 1)
For k = 2:
P(X = 2) = C(10, 2) * 0.49^2 * (1 - 0.49)^(10 - 2)
For k = 3:
P(X = 3) = C(10, 3) * 0.49^3 * (1 - 0.49)^(10 - 3)
Finally, we can sum up all these probabilities to get the probability that at most 3 commuters listen to audiobooks:
P(at most 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
Now, you can calculate each of these probabilities and find the sum to get the final answer.