I have a bag full of 10 marbles: 2 black, 1 blue, 1 yellow, 3 green, 1 brown, and 2 purple. I draw 5 marbles one at a time without replacement. What is the probability of a black marble to be in the five drawn?
[Math] Probability without replacement question
probability
Best Answer
Think of the marbles as having, in addition to colour, an ID number that makes them distinct.
There are two interpretations of "one black:" A: at least one black, and B: exactly one black. The probabilities are of course different. My preferred interpretation of the wording is A. Edit: With the change of wording to "a black" it is clearly A that is meant, but for your interest I will keep the analysis of B.
A: At least one black: It is easier to find first the probability of no black.
There are $\binom{10}{5}$ ways to choose $5$ marbles, all equally likely. Note that there are $\binom{8}{5}$ ways to choose $5$ marbles from the $8$ non-black. So the probability that all the balls are non-black is $$\frac{\binom{8}{5}}{\binom{10}{5}},$$ and therefore the probability of at least one black is $$1-\frac{\binom{8}{5}}{\binom{10}{5}}.$$
B: Exactly one black: There are $\binom{2}{1}$ ways of choosing one black from the two available. For each such way, there are $\binom{8}{4}$ ways to choose the non-blacks to go with it. So the total number of ways to pick exactly one black, and the rest non-black, is $\binom{2}{1}\binom{8}{4}$. Thus our probability is $$\frac{\binom{2}{1}\binom{8}{4}}{\binom{10}{5}}.$$
Remark: We used general techniques. For A, there is a simpler way. The probability that the first marble chosen is non-black is $\frac{8}{10}$. Given the first was non-black, the probability that the second is non-black is $\frac{7}{9}$, since there are $7$ non-blacks left in a total of $9$. So the probability the first two are non-black is $\frac{8}{10}\cdot\frac{7}{9}$. Continue in this way. The probability all five are non-black is $$\frac{8}{10}\cdot\frac{7}{9}\cdot\frac{6}{8}\cdot\frac{5}{7}\cdot\frac{4}{6}.$$ As in the earlier discussion, subtract the above from $1$.