Question

(Extra credit) All human blood can be typed as one of O, A, B, or AB. The distribution of the type varies a bit with race. For African-Americans, here are the approximate probabilities that a person will have blood type O, A, B, or AB. Blood Type O A B AB Probability 0.4 0.2 0.32 0.08 In a group of 10 randomly chosen people, what is the probability that two or more of them have Type A blood

Answer 1

Probability that two or more of them have Type A blood is 0.6242.

Explanation:

We are given the approximate probabilities that a person will have blood type O, A, B, or AB.

Blood Type O A B AB

Probability 0.4 0.2 0.32 0.08

A group of 10 people are chosen randomly.

The above situation can be represented through Binomial distribution;

`P(X=r) = \binom{n}{r}p^(r) (1-p)^(n-r) ; x = 0,1,2,3,.....`

where, n = number of trials (samples) taken = 10 people

r = number of success = two or more have Type A blood

p = probability of success which in our question is probability

that a person has Type A Blood, i.e; p = 20% or 0.20

LET X = Number of person having Type A Blood

So, it means X ~ Binom(n = 10, p = 0.20)

Now, Probability that two or more of them have Type A blood is given by = P(X
`\geq` 2)

P(X
`\geq` 2) = 1 - P(X = 0) - P(X = 1)

=
`1- \binom{10}{0}* 0.20^(0) * (1-0.20)^(10-0)-\binom{10}{1}* 0.20^(1) * (1-0.20)^(10-1)`

=
`1- 1 * 1 * 0.80^(10)-10 * 0.20 * 0.80^(9)`

= 0.6242

Hence, the probability that two or more of them have Type A blood is 0.6242.