a) i) Since blood types across individuals are independent, we have
Pr[1 = A and 2 = B] = Pr[1 = A] • Pr[2 = B]
where e.g. Pr[1 = A] means "probability that partner 1 has type A blood". According to the given distribution,
Pr[1 = A and 2 = B] = 0.37 • 0.13 = 0.0481 = 4.81%
a) ii) The event that partner 1 has type A and partner 2 has type B is exclusive from the event that partner 1 has type B and partner 2 has type A, and vice versa. In other words, only one of these events can happen, and not both simultaneously. This means the probability of either event occurring is equal to the sum of the probabilities of these events occurring individually. Then
Pr[(1 = A and 2 = B) or (1 = B and 2 = A)]
= Pr[1 = A and 2 = B] + Pr[1 = B and 2 = A]
Both probabilities are the same, and we found it in the previous part. So
Pr[(1 = A and 2 = B) or (1 = B and 2 = A)]
= 0.0481 + 0.0481 = 0.0962 = 9.62%
a) iii) The event that at least one partner has type O occurs if exactly one partner has type O, or both do. That is, "at least one partner with type O" contains 3 possible events,
• 1 = O and 2 = not O
• 1 = not O and 2 = O
• 1 = O and 2 = O
For the first event, by independence, is
Pr[1 = O and 2 = not O] = Pr[1 = O] • Pr[2 = not O]
and by definition of complementary event,
Pr[1 = O and 2 = not O] = Pr[1 = O] • (1 - Pr[2 = O])
From the given distribution, it follows that
Pr[1 = O and 2 = not O] = 0.44 • (1 - 0.44) = 0.2464 = 24.64%
The second event is the same as the first in terms of probability.
For the third event, we have
Pr[1 = O and 2 = O] = Pr[1 = O] • Pr[2 = O] = 0.44² = 0.1936 = 19.36%
Then the total probability of at least one O-type partner is
24.64% + 24.64% + 19.36% = 68.64%
b) Suppose partner 1 is type-B. Then partner 2 is an acceptable donor if they are either type-B or type-O. So we want to find
Pr[1 = O and (2 = B or 2 = O)]
By independence,
= Pr[1 = O] • Pr[2 = B or 2 = O]
By mutual exclusivity,
= Pr[1 = O] • (Pr[2 = B] + Pr[2 = O])
Then from the given distribution,
= 0.44 • (0.13 + 0.44) = 0.2508 = 25.08%
Now suppose partner 2 is type-B. Then partner 1 must be either type-B or type-O. But the math works out the same way, so that the overall probability is
25.08% + 25.08% = 50.16%
