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:
1) The probability that at least 18 customers lost baggage is____?
A specific answer would be awesome, because I'm not that bright when it comes to scientific calculators.. thanks.
same question as yours ...
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To find the probability that at least 18 customers lost their baggage, we can use the binomial probability formula. The formula is:
P(x) = C(n, x) * p^x * q^(n-x),
where:
P(x) is the probability of exactly x successes,
C(n, x) is the combination function "n choose x",
p is the probability of success for each customer (8% or 0.08),
q is the probability of failure (1 - p), and
n is the total number of customers (20).
In this case, we want to find the probability of "x >= 18," which means the probability of 18, 19, or 20 customers losing their bags.
We can calculate the probability for each case and add them together. Let's go step by step:
For x = 18:
P(18) = C(20, 18) * (0.08)^18 * (0.92)^2.
Using the combination function, we have:
C(20, 18) = 20! / (18! * (20-18)!) = 20! / (18! * 2!) = 190.
Calculating further:
P(18) = 190 * (0.08)^18 * (0.92)^2 ≈ 0.005127036.
For x = 19:
P(19) = C(20, 19) * (0.08)^19 * (0.92)^1.
Using the combination function, we have:
C(20, 19) = 20! / (19! * (20-19)!) = 20! / (19! * 1!) = 20.
Calculating further:
P(19) = 20 * (0.08)^19 * (0.92)^1 ≈ 0.000254871.
For x = 20:
P(20) = C(20, 20) * (0.08)^20 * (0.92)^0.
Using the combination function, we have:
C(20, 20) = 20! / (20! * (20-20)!) = 20! / (20! * 0!) = 1.
To calculate further:
P(20) = 1 * (0.08)^20 * (0.92)^0 ≈ 2.56e-8.
Finally, we add the probabilities together:
P(x >= 18) = P(18) + P(19) + P(20) ≈ 0.005127036 + 0.000254871 + 2.56e-8 ≈ 0.005381301.
Therefore, the probability that at least 18 customers lost their baggage is approximately 0.005381301 (rounded to 6 decimal places).