Customers lose their bags 8% of the time at airports.

20 random people were watched to see if their bags were lost. Whats the probability that at least 18 of them lost their bags?

at least 18 lost their bags

implies
18, 19 or 20 lost their bags.

prob of losing bag = .08
prob of not losing bag = .92

prob as stated
= C(20,18)(.08)^18 (.92)^2 + C(20,19)(.08)^19 (.92) + .08^20
= ....

you do the button-pushing

To find the probability that at least 18 out of 20 random people lose their bags, 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 exactly k successes,
C(n, k) is the combination formula for selecting k items from a set of n,
p is the probability of a single success, and
n is the number of trials.

In this case, we want to know the probability of at least 18 successes, so we need to calculate the probability for 18, 19, and 20 successes and sum them together.

Let's plug in the given values into the binomial probability formula:

p = 0.08 (The probability of losing a bag = 8% = 0.08)
n = 20 (Number of random people watched)

Now let's calculate the probabilities for 18, 19, and 20 successes (people losing their bags) and sum them together.

P(X = 18) = C(20, 18) * 0.08^18 * (1-0.08)^(20-18)
P(X = 19) = C(20, 19) * 0.08^19 * (1-0.08)^(20-19)
P(X = 20) = C(20, 20) * 0.08^20 * (1-0.08)^(20-20)

To calculate the combination formula C(n, k), use the formula:
C(n, k) = n! / (k! * (n-k)!), where ! represents factorial.

Now, let's calculate each probability:

C(20, 18) = 20! / (18! * (20-18)!) = 20! / (18! * 2!) = (20 * 19) / 2 = 190
C(20, 19) = 20! / (19! * (20-19)!) = 20! / (19! * 1!) = (20 * 1) / 1 = 20
C(20, 20) = 20! / (20! * (20-20)!) = 20! / (20! * 0!) = 1

Now, let's calculate the probabilities:

P(X = 18) = 190 * 0.08^18 * 0.92^2
P(X = 19) = 20 * 0.08^19 * 0.92^1
P(X = 20) = 1 * 0.08^20 * 0.92^0

Finally, let's find the probability of at least 18 successes by summing the three probabilities:

P(X >= 18) = P(X = 18) + P(X = 19) + P(X = 20)

Calculate the individual probabilities and sum them up to find the final probability.