Suppose there are 5 M and M beans of different colors. You randomly choose 3. What is the probability that you guess 2 out of the 3 numbers correctly?
Apparently the answer is $12/125$, but I don't see where this came from.
Here is my take on this. Consider the 3 selected beans. What is the probability that you guess 2 out of the 3 correctly? To guess 2 correctly, there are $\binom{3}{2}$ different pairs you can guess correctly. For each pair, there is a $\frac{1}{5} \cdot \frac{1}{4} \cdot \frac{2}{3}$ probability of guessing it correctly. So I believe the answer should be
$$ \binom{3}{2} \frac{1}{5} \cdot \frac{1}{4} \cdot \frac{2}{3} = \frac{1}{10} $$
Am I thinking about this correctly?
$\endgroup$ 01 Answer
$\begingroup$In both answers, the order of the candies appears to be relevant. The difference between the answers has to do with the strategy for guessing. In your answer, you are not repeating your guesses. In the given answer the guesses are made independently and guesses may be repeated (despite this lowering the probability, as shown).
Note $\frac{12}{125}=3\cdot \frac{1}{5}\cdot \frac{1}{5}\cdot \frac{4}{5}$
Which answer is "more correct?" Without more context, I would have leaned towards your answer myself as it is commonly assumed that such guessing would be done in an optimal way that maximizes chances of success.
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