33 lines
666 B
Markdown
33 lines
666 B
Markdown
#Math #Probability
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# Problem
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Given $m$ items of one type and $n$ items of another type, what is the probability of choosing $l$ items of type one and $o$ items of type two if you pick $l + o$ items?
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# Solution
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Total ways to choose the items not considering types:
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$$
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{m + n} \choose {l + o}
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$$
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Total ways to choose $l$ items of type one:
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$$
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m \choose l
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$$
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Total ways to choose $o$ items of type two:
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$$
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n \choose o
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$$
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Multiply the ways to choose both items to get the number of ways to choose $l$ items of type one and $o$ items of type two, divide by total number of combinations:
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$$
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\frac {{m \choose l} {n \choose o}} {{m + n} \choose {l + o}}
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$$ |