SOLUTION: A courier service promises that 80% of deliveries will reach their destinations within 12 hours.what is the probability that of the 7 parcels sent at random times only one is del

Question 1074311: A courier service promises that 80% of deliveries will reach their destinations within 12 hours.what is the probability that of the 7 parcels sent at random times only one is delivered late?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
binomial formula is p(x) = p^x * q^(n-x) * c(n,x)

p is the probability of occurrence.
p is the probability of no occurrence = 1 - p.
x is the number of occurrences.
(n-x) is the number of non-occurrences.
c(n,x) is the combination formula for number of ways you can get x out of n without regard to order.

the probability of occurrence here would be the probability that the package is being delivered late.

since 80% arrive within 12 hours, then 20% must not arrive within 12 hours.

therefore, p = .2 and q = .8

p means the probability the package will be late.
q means the probability the paqckage will be on time.

if at least 6 out of 7 arrive on time, this means that at most 1 arrives late.

this means that either 0 arrive later or 1 arrives late.

therefore, you want p(0) + p(1).

p(0) = .2^0 * .8^7 * c(7,0) = 1 * .2097152 * 1 = .2097151
p(1) = .2^1 * .8^6 * c(7,1) = .2 * .262144 * 7 = .3670016

the total probability of p(0) + p(1) = .5767526

the following pictures shows all the probabilities from x = 0 to x = 7

their sum has to be equal to 1, as it is.

$$$

the numbers shown for p(1) and p(2) are rounded a little more than the numbers i showed you above, but they are the same numbers.

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