Losing your bags during airline travel is not all that uncommon. Recently, an airline was sited for losing customers' baggage 8% of the time.

A test is conducted by randomly selecting 14 customers for that airline and observing whether they had lost their baggage. Provide answers to at least 6 decimal places for the following:

(a) The probability that exactly 2 customers lost baggage is .

(b) The probability that at least 12 customers lost baggage is

prob of losing = 8/100 = 2/25

prob of not losing it = 23/25

prob(2 out of 14losing it) = C(14,2) (2/25)^2 (23/25)^12 = appr .214

b) prob (at least 12)
= Prob(12) + prob(13) + prob(14)
= C(14,12) (2/25)^12 (23/25)^2 + .....

I do not understand the second part how do i get the second part?

Doesn't the second part yield a value greater than 1? Isn't probability always supposed to equal 1?

i have no clue honestly i tried doing the second part B and i do not get it?

To find the probabilities in this scenario, we can use the binomial probability formula:

P(X = k) = (n C k) * p^k * (1-p)^(n-k)

Where:
P(X = k) is the probability of getting exactly k successes,
n is the total number of trials or customers in this case,
p is the probability of success, and
(n C k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)

(a) The probability that exactly 2 customers lost baggage is:

Let's substitute the given values into the formula:

n = 14 (total number of customers)
k = 2 (number of customers losing baggage)
p = 0.08 (probability of losing baggage)

P(X = 2) = (14 C 2) * 0.08^2 * (1-0.08)^(14-2)

Calculating the probability:

P(X = 2) = (14! / (2! * (14-2)!) * 0.08^2 * (1-0.08)^12
P(X = 2) = (14! / (2! * 12!)) * 0.08^2 * 0.92^12

Using a calculator or software, you can compute the value of (14! / (2! * 12!)) to find the binomial coefficient, which is 91.

P(X = 2) = 91 * 0.08^2 * 0.92^12

Calculating further:

P(X = 2) ≈ 0.198641

So, the probability that exactly 2 customers lost baggage is approximately 0.198641.

(b) The probability that at least 12 customers lost baggage is:

To find this probability, we need to calculate the sum of probabilities for 12, 13, and 14 customers losing their baggage.

P(X ≥ 12) = P(X = 12) + P(X = 13) + P(X = 14)

We can substitute the values for n, p, and use the same formula as above to find the individual probabilities:

P(X ≥ 12) = (14 C 12) * 0.08^12 * (1-0.08)^(14-12) + (14 C 13) * 0.08^13 * (1-0.08)^(14-13) + (14 C 14) * 0.08^14 * (1-0.08)^(14-14)

Calculating these probabilities individually, sum them up to find the final result.

Note: The binomial coefficient (14 C k) can be calculated as (14! / (k! * (14-k)!)).

Using a calculator or software can help calculate these expressions more precisely.