Question 1151341: Suppose that 20% of the population has type A blood.
(a) If 8 people are selected at random, what is the probability that less than 3 have type A blood.
(b) If 80 donors come to give blood one day, what is the probability that less than thirty of them have Type A blood (using the normal approximation)? Explain why this is higher or lower than the answer in part (a).
(c) If 20 people come to give blood, what is the probability that at least one of the donors is of Type A? (I’d be impressed if you could solve this in two different ways. . . )
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! fewer than 3 is 0,1,2
for 0 this is 1*0.2^0*0.8^8=0.1678
for 1 this is 8*0.2*0.8^7=0.3355
for 2 this is 8C2*0.2^2*0.8^6=0.2936
0.7969
mean is 80*.20 or 16 np and p is 0.2
variance is np(1-p) or 16*.8=12.8
sd is sqrt(V)=3.58
fewer than 30 is using continuity correction factor is z<(29.5-16)/3.58 or <3.77 or essentially 100%.
As the sample size increases, the error from the mean decreases. It is quite likely to have outliers with a small sample; it is far less likely (1/ sqrt(n)) to have the same error with a larger sample size.
what is the probability none is type A? That is 0.8^20=0.0115
The complement, 1-0.0115 or 0.9885 is the probability at least one is type A.
normal approximation: mean is 20*0.2 or 4
V is 4(0.8)=3.2
sd is sqrt(3.2)=1.79
for at least 1 what is probability z> (0.5-4) / 1.79 (continuity correction factor)
or z>-3.5/1.79 or 0.9747
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