Question

Two t-shirts are drawn at random in succession without replacement from a drawer containing 5 red t-shirts and 8 white t-shirts. Find the probabilities of all the possible outcomes.

(a) 1

(b) 13

(c) 40

(d) 346

This question was posed to me in a job interview.

The origin of the question is Discrete Probability topic in portion Discrete Probability of Discrete Mathematics

Answer 1

Correct choice is (a) 1

Explanation: Let X denote the number of red t-shirts in the outcome. Here, x1 = 2, x2 = 1, x3 = 1, x4 = 1, x5 = 0. Probability of first t-shirt being red = \(\frac{5}{13}\).

Probability of second t-shirt being red = \(\frac{4}{12}\).

So: P(x1) = \(\frac{5}{13} × \frac{4}{12} = \frac{20}{146}\). Likewise, for the probability of red first followed by black is \(\frac{8}{12}\) (as there are 8 red t-shirts still in the drawer and 12 t-shirts all together).

So, P(x2) = \(\frac{5}{13} *\frac{8}{12} = \frac{40}{146}\). Similarly for white then red: P(x3) = \(\frac{8}{13} × \frac{4}{12} = \frac{32}{146}\).  Finally, for 2 black balls: P(x4) = \(\frac{8}{13} × \frac{7}{12} = \frac{56}{146}\). So, \(\frac{20}{146} + \frac{40}{146} + \frac{32}{146} + \frac{40}{146} = 1\). Hence, all the t-shirts have been found.