Question

Four people \(-\mathrm{a}, \mathrm{b}, \mathrm{c}\), and \(\mathrm{d}-\) are waiting to give blood. Of these four, a and b have type AB blood, whereas c and d do not. An emergency call has just come in for some type AB blood. If blood samples are taken one by one from the four people in random order for blood typing and \(x\) is the number of samples taken to obtain an \(\mathrm{AB}\) individual (so possible \(x\) values are 1,2 , and 3 ), what is the probability distribution of \(x\) ?

Short answer

The probability distribution of \(x\) is: \(P(x = 1) = 0.5\), \(P(x = 2) = \frac{1}{3}\), \(P(x = 3) = \frac{1}{6}\)

Step-by-step solution

Determine the total number of individuals

From the problem, we know that there are 4 individuals: a, b, c and d. Out of these 4, two (a and b) have type AB blood. So the total number of individuals is 4.

Calculate the probability for \(x = 1\)

If \(x = 1\), this means that the first sample taken is from an AB individual. There are two AB individuals out of the total four. So probability of selecting an AB individual in the first attempt is \(\frac{2}{4} = 0.5\)

Calculate the probability for \(x = 2\)

If \(x = 2\), this means that the first sample was not from an AB individual, but the second one is. The first person was either c or d (not-AB), and the second one was either a or b (AB). So the probability is the product of the probability of not selecting an AB individual in the first attempt and selecting an AB individual in the second attempt. So \(P(x = 2) = \frac{2}{4} \times \frac{2}{3} = \frac{1}{3}\)

Calculate the probability for \(x = 3\)

If \(x = 3\), this means that the first and second samples were not from AB individuals while the third one was. The probability for this scenario is calculated by multiplying the probabilities of: not selecting an AB individual in the first attempt, not selecting an remaining AB individual in the second attempt, and selecting an remaining AB individual in the third attempt. So \(P(x = 3) = \frac{2}{4} \times \frac{1}{3} \times \frac{2}{2} = \frac{1}{6}\)