transportation officials reported that 8.25 out every 1000 airline passengers lost luggage during their travels last year.if we randomly select 400 airline passengers,what is the probability that 5 lost some luggage?
To find the probability that 5 out of 400 airline passengers lost some luggage, we can use the binomial probability formula. The formula is:
P(x) = (nCx) * (p^x) * ((1-p)^(n-x))
Where:
- P(x) is the probability of getting exactly x successes
- n is the total number of trials or passengers (in this case, 400)
- x is the number of successful outcomes or passengers who lost luggage (in this case, 5)
- p is the probability of success (in this case, the probability of losing luggage per passenger)
In this case, the probability of losing luggage per passenger is given as 8.25 out of every 1000 passengers. We can convert this to a probability by dividing it by 1000:
p = 8.25 / 1000 = 0.00825
Now we can substitute the values into the formula:
P(5) = (400C5) * (0.00825^5) * ((1-0.00825)^(400-5))
To calculate this, we need to use the combination formula for (nCx), which is:
nCx = n! / (x! * (n-x)!)
Calculating each part step by step:
- (400C5) = 400! / (5! * (400-5)!)
- (0.00825^5) = 0.00825^5
- ((1-0.00825)^(400-5)) = (1-0.00825)^(400-5)
Calculating the combination:
(400C5) = 400! / (5! * (400-5)!) = (400 * 399 * 398 * 397 * 396) / (5 * 4 * 3 * 2 * 1)
Now, we can substitute these values into the binomial probability formula and calculate the result:
P(5) = [(400 * 399 * 398 * 397 * 396) / (5 * 4 * 3 * 2 * 1)] * (0.00825^5) * ((1-0.00825)^(400-5))
The result will be the probability that exactly 5 out of 400 airline passengers lost some luggage.