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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