[Math] Find the following probability

probabilityprobability theory

A bowl contains 16 chips, of which 6 are red, 7 are white and 3 are blue. If four chips are taken at random and without replacement, find the probability that there is at least 1 chip of each colour.

Can someone please give me a hint?

Thank you!

Best Answer

I think barak manos is doubly counting some selections. Following Marconius's tip, we have

\begin{align} \binom{6}{2}(7)(3) + (6)\binom{7}{2}(3) + (6)(7)\binom{3}{2} & = (15)(7)(3) + (6)(21)(3) + (6)(7)(3) \\ & = 315 + 378 + 126 = 819 \end{align}

different ways to select chips to satisfy the condition. Note that there are

$$ \binom{6}{4} + \binom{7}{4} = 15 + 35 = 50 $$

different ways to select only red chips or only white chips, and that is the difference between this answer and barak's.

As in barak's answer, there are

$$ \binom{16}{4} = 1820 $$

different ways to select $4$ of $16$ chips, so the desired probability is

$$ P = \frac{819}{1820} = \frac{9}{20} $$

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