First, calculate the probability of a person being group O+ given they are Rhesus +. The probability of being O+ is 37%, and the probability of being Rhesus + is the sum of A+, B+, AB+, and O+, which is 35% + 8% + 3% + 37% = 83%.
Thus, the probability of being O+ given Rhesus + is:
\(P(O \text{ given } +) = \frac{0.37}{0.83} = 0.4458\)
Now, use the binomial distribution to find the probability that fewer than 3 out of 9 people are group O+.
Let \(X\) be the number of people who are group O+. Then \(X \sim \text{Binomial}(9, 0.4458)\).
We need to find \(P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)\).
\(P(X = 0) = \binom{9}{0} (0.4458)^0 (0.5542)^9\)
\(P(X = 1) = \binom{9}{1} (0.4458)^1 (0.5542)^8\)
\(P(X = 2) = \binom{9}{2} (0.4458)^2 (0.5542)^7\)
Calculating these:
\(P(X = 0) = (0.5542)^9\)
\(P(X = 1) = 9 \times (0.4458) \times (0.5542)^8\)
\(P(X = 2) = 36 \times (0.4458)^2 \times (0.5542)^7\)
Summing these probabilities gives:
\(P(X < 3) = 0.156\)