Probability: Harder Probability - Using Combinations
The chance of rain on any given day in Tel-Aviv is 50%. What is the probability that it rained in Tel-Aviv on exactly 4 out of 6 randomly chosen days?
Correct.
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Find the number of possible ways to pick 4 days out of 6. Order of choice doesn't matter because you only care about which days are chosen, not the order you chose them in. If, for example, you choose the first 4 days to be rainy, it does not matter in what order you chose them. Now that we have reduced the problem to a simple case of choosing $$\color{green}{k=4}$$ out of $$\color{red}{n=6}$$, __order doesn't matter__. Use the __Combinations__ formula:
> $$C(n{,}k) = \frac{n!}{(n-k)!k!} = \frac{6!}{(6-4)!4!} = \frac{6!}{2!4!} = \frac{6\times5}{2} =15$$.
So the probability that you're looking for is the __number of arrangements__ times the __probability__ of each arrangement:
>$$\text{Probability} = 15\times \frac{1}{64} = \frac{15}{64}$$.
Incorrect.
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$$\frac{15}{64}$$
$$\frac{1}{4}$$
$$\frac{15}{32}$$
$$\frac{1}{2}$$
$$\frac{13}{16}$$
