Losing your bags during airline travel is not all that uncommon. Recently, an airline was cited for losing customers' baggage 8% of the time.
A test is conducted by randomly selecting 20 customers for that airline and observing whether they had lost their baggage. Provide answers to at least 6 decimal places for the following:
The probability that at least 16 customers lost baggage is
To find the probability that at least 16 customers lost baggage, we can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (nCx) * p^x * (1-p)^(n-x),
where:
- P(x) is the probability of x successes,
- n is the total number of trials,
- p is the probability of success in each trial,
- nCx is the number of ways to choose x successes from n trials, and
- (1-p)^(n-x) is the probability of failure in each of the remaining trials.
In this case, the probability of losing baggage is given as 8% or 0.08, and we have 20 customers selected randomly.
To find the probability that at least 16 customers lost baggage, we need to calculate the sum of the individual probabilities for 16, 17, 18, 19, and 20 successes.
P(at least 16 customers losing baggage) = P(16) + P(17) + P(18) + P(19) + P(20)
We can plug in the values into the binomial probability formula to calculate each individual probability:
P(16) = (20C16) * (0.08)^16 * (1-0.08)^(20-16)
P(17) = (20C17) * (0.08)^17 * (1-0.08)^(20-17)
P(18) = (20C18) * (0.08)^18 * (1-0.08)^(20-18)
P(19) = (20C19) * (0.08)^19 * (1-0.08)^(20-19)
P(20) = (20C20) * (0.08)^20 * (1-0.08)^(20-20)
To calculate this, we need to use combination formula:
nCx = n! / (x! * (n-x)!)
Let's calculate these probabilities step by step:
P(16) = (20C16) * (0.08)^16 * (1-0.08)^(20-16)
= (20! / (16! * (20-16)!)) * (0.08)^16 * (0.92)^4
Calculating this value will give us the probability of having exactly 16 customers losing their baggage.
Repeat this process for probabilities P(17), P(18), P(19), and P(20).
After calculating the individual probabilities, sum them up to find the probability that at least 16 customers lost baggage:
P(at least 16 customers losing baggage) = P(16) + P(17) + P(18) + P(19) + P(20)
Make sure to round the final answer to at least 6 decimal places.