Answer: Option B is the only correct option.
Step-by-step explanation:
Number of samples = n = 8.
Probability of success = p = 0.4
Probability of failure = q = 0.6
r = chosen number of donors among the 8
To solve this question, we use the distribution formula
P(x=r) = nCr * p^r * q^n-r
For option A, to check if P(3<x<5) = 0.37. [3 and 5 inclusive]
When x = 3
P(x=3) = 8C3 * 0.4^3 * 0.6^5
P(x=3) = 56 * 0.064 * 0.07776
P(x=3) = 0.2787
When x= 4
P(x=4) = 8C4 * 0.4^4 * 0.6^4
P(x=4) = 70 * 0.0256 * 0.1296
P(x=4) = 0.2322
Since p(x=3) + p(x=4) is already greater than 0.37, then we know option A is NOT correct.
For option B, To check if the probability of 1 or fewer donor is about 0.11. i.e if P(x</=1) = 0.11
When x=o
P(x=0) = 8C0 * 0.4^0 * 0.6^8
P(x=0) = 1* 1 * 0.016796
P(x=0) = 0.016796.
When x = 1
P(x=1) = 8C1 * 0.4^1 * 0.6^7
P(x=1) = 8 * 0.4 * 0.02799
P(x=1) = 0.08958
P(x=0) + P(x=1) = 0.016796 + 0.08958
P(x=0) + P(x=1) = 0.10635.
Since this is approximately 0.11, then option B is a correct option.
For option C to check if the probability 7 or more donors not having type A = 0.0087
To do this,we determine thw probability of 7 or more donors having type A and we subtract our answer from 1.
First, we determine P(x>/=7)
When x= 7
P(x=7) = 8C7 * 0.4^7 * 0.6^1
P(x=7) = 8 * 0.001638 * 0.6
P(x=7) = 0.007864
When x=8
P(x=8) = 8C8 * 0.4^8 * 0.6^0
P(x=8) = 1 * 0.0006554 * 1
P(x=8) = 0.0006554
P(x=7) + P(x=8) = 0.007864 + 0.0006554 = 0.00852.
Since probability of 7 or more donors having type A is 0.00852 as against what was stated in the option C, then option C is NOT a correct option.
For option D, to check if the probability of exactly 5donors having type A blood = 0.28
When x=5
P(x=5) = 8C5 * 0.4^5 * 0.6^3
P(x=5) = 56 * 0.01024 * 0.216
P(x=5) = 0.1239.
Since probability of what was derived for having exactly 5 donors having sample A is different from what wqs given in the option, then option D is NOT correct.
For option E, since what was stated in the option negates what was derived for exactly 5 donors, then option E is NOT correct
