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.
To solve this problem, we can use the binomial probability formula.
The formula for the probability of getting exactly "x" successes in "n" trials is:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
- n is the total number of trials
- x is the number of successful trials
- p is the probability of success in one trial
In this case, we have n = 10 (sample size), x can be 0, 1, 2, or 3 (at most 3), and p = 0.49 (probability of a commuter listening to audiobooks).
Now we can calculate the probability using the formula for each value of x and add them up.
P(0) = (10C0) * 0.49^0 * 0.51^10
P(1) = (10C1) * 0.49^1 * 0.51^9
P(2) = (10C2) * 0.49^2 * 0.51^8
P(3) = (10C3) * 0.49^3 * 0.51^7
Adding up these individual probabilities will give us the probability of at most 3 commuters listening to audiobooks.
P(at most 3) = P(0) + P(1) + P(2) + P(3)
Calculating this:
P(at most 3) = (10C0) * 0.49^0 * 0.51^10 + (10C1) * 0.49^1 * 0.51^9 + (10C2) * 0.49^2 * 0.51^8 + (10C3) * 0.49^3 * 0.51^7
P(at most 3) ≈ 0.10
Therefore, the probability that at most 3 commuters listen to audiobooks is approximately 0.10.
To find the probability that at most 3 commuters listen to audiobooks, we need to calculate the probability of 0, 1, 2, and 3 commuters listening to audiobooks, and then sum all those probabilities.
First, let's calculate the probability of 0 commuters listening to audiobooks.
The probability of one commuter not listening to audiobooks is 1 - 0.49 = 0.51.
Since we have 10 commuters in the sample, we need to calculate (0.51)^10.
P(0 commuters) = (0.51)^10 = 0.00147825634
Next, let's calculate the probability of 1 commuter listening to audiobooks.
The probability of one commuter listening to audiobooks is 0.49.
The probability of the other 9 commuters not listening to audiobooks is (0.51)^9.
P(1 commuter) = (0.49)*(0.51)^9 = 0.0142878769
Then, let's calculate the probability of 2 commuters listening to audiobooks.
The probability of two commuters listening to audiobooks is (0.49)^2.
The probability of the other 8 commuters not listening to audiobooks is (0.51)^8.
P(2 commuters) = (0.49)^2 * (0.51)^8 = 0.065883214
Finally, let's calculate the probability of 3 commuters listening to audiobooks.
The probability of three commuters listening to audiobooks is (0.49)^3.
The probability of the other 7 commuters not listening to audiobooks is (0.51)^7.
P(3 commuters) = (0.49)^3 * (0.51)^7 = 0.1809473804
Now, we sum the probabilities:
P(at most 3 commuters) = P(0 commuters) + P(1 commuter) + P(2 commuters) + P(3 commuters)
P(at most 3 commuters) = 0.00147825634 + 0.0142878769 + 0.065883214 + 0.1809473804
Calculating this sum gives us the answer:
P(at most 3 commuters) ≈ 0.2626
Therefore, the probability that at most 3 of them listen to audiobooks is approximately 0.26.