Balls of 3 colours in a bag - probability of getting 3 same color balls in 3 out of 5 draws

Question

There are balls of 3 colours in a bag: 3 red, 4 green, 5 blue. Randomly perform 5 draws. In each draw, retrieve 3 balls from the bag at the same time . And place the balls back and perform next draw. What is the probability of getting 3 balls of same color in any 3 draws (means excluding 1,2,4,5 draws of the same color)?

Answer 1

Stipulation: What we want, I'm assuming, is the probability that out of $5$ three-ball draws, exactly $3$ draws have all three balls with the same color (either red or green or blue).

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First, find the probability that any given draw consists of just one color.

\begin{align} P(\text{all one color}) & = P(\text{all red}) + P(\text{all green}) + P(\text{all blue}) \\ & = \frac{\binom{3}{3}}{\binom{12}{3}} + \frac{\binom{4}{3}}{\binom{12}{3}} + \frac{\binom{5}{3}}{\binom{12}{3}} \end{align}

Let this quantity be denoted $q$. Then you can just use the binomial theorem to determine the probability that exactly three of these three-ball draws end up with all one color:

$$ P(\text{exactly $3$ of the $5$ three-ball draws are all one color}) = \binom{5}{3} q^3 (1-q)^2 $$