Only about 75% of all donated human blood can be used in hospitals. the remaining 25% cannot be used because of various infections in the blood. suppose a blood bank has 10 newly donated pints of blood. let r be a binomial random variable that represents the number of "good" pints that can be used.
(a) based on questionnaires completed by the donors, it is believed that at least 6 of the 10 pints are usable. what is the probability that at least 8 of the pints are usable, given this belief is true? compute p(8 ≤ r | 6 ≤ r). (round your answer to three decimal places.)
(b) assuming the belief that at least 6 of the pints are usable is true, what is the probability that all 10 pints can be used? compute p(r = 10 | 6 ≤ r). (round your answer to three decimal places.)

a) 0.02739

b) 0.00034

Step-by-step explanation:

If it is believed that at least 6 of the 10 pints are usable, then the 75% of “good” pints that can be used from this sample reduces to 60% of 75% = 0.6*0.75 = 0.45

Now for this sample we use a binomial distribution with probability of “success” (finding a “good” pint) of 0.45% and

the probability of getting exactly r good pints out of 10 is  

where

are combinations of 10 taken r at a time.

a)

The probability that at least 8 of the pints are usable is P(r>7)

b)

Here we want P(r=10)

the simplified form is

step-by-step explanation:

we want to simplify the expression,

let us change the middle bar to a normal division sign to obtain,

we multiply the first fraction by the reciprocal of the second fraction to obtain,

we factor to obtain,

we cancel the common factors to get,

we simplify to get,

the correct answer is b