Question

A basket contains 3 oranges and 3 mangoes. A playful baby picks two fruits at random from this basket. If X denotes the number of oranges thus picked, determine the probability distribution of X

Answer 1

There are
`\binom62=15` ways of selecting any two fruit from the basket, where

`\dbinom nk = (n!)/(k!(n-k)!)`

is the so-called binomial coefficient.

If the baby picks out
`k` oranges, where
`k\in\{0,1,2\}`, it can do so in

`\dbinom 3k \dbinom3{2-k} = (3!)/(k!) (3!)/((2-k)! (1 + k)!)`

ways.

Then the PMF (probability mass function) of
`X` is

`P(X=x) = \begin{cases} (\binom30 \binom32)/(\binom62) = \frac3{15} & \text{if }x=0 \\\\ (\binom31 \binom31)/(\binom62) = \frac9{15} & \text{if }x = 1 \\\\ (\binom32 \binom30)/(\binom62) = \frac3{15} & \text{if }x=2 \\\\ 0 &\text{otherwise}\end{cases}`