How Is the Probability Mass Function Calculated for Identifying Blood Type A+?

[war485]

Homework Statement

There is a group of four blood donors: A, B, C and D. Only A has type
A+ blood. Four blood samples, one from each individual, will be typed in random order until an A+ individual is identified. Let Y = {number of typings necessary to identify an A+ individual}. Compute the probability mass function.

Homework Equations

permutation (order matters): n! / (n - k)!

The Attempt at a Solution

let p = probability
p(Y=1) = 1/4
p(Y=2) = permute (1,1) * permute (1,3) / permute (2,4) = 1/4
p(Y=3) = permute (1,1) * permute (2,3) / permute (3,4) = 1/4
p(Y=4) = permute (1,1) * permute (3,3) / permute (4,4) = 1/4

Even though I think the specific order matters, it doesn't make sense that they each have a probability of 1/4. Did I miss something from the problem? Does this seem right/logical?

 

[Random Variable]

Seems right to me.

$$P(Y=1) = \frac {1}{4}$$

$$P(Y=2) = (\frac {3}{4}) (\frac {1}{3}) = \frac {3}{12} = \frac {1}{4}$$

$$P(Y=3) = (\frac {3}{4}) (\frac {2}{3}) (\frac {1}{2}) = \frac {6}{24} = \frac {1}{4}$$

$$P(Y=4) = (\frac {3}{4}) (\frac {2}{3}) (\frac {1}{2}) (\frac{1}{1})= \frac {6}{24} = \frac {1}{4}$$