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August 27th, 2018, 06:37 AM   #1
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Differing probabilities .. stuck need help

I have tried to work this out but my answers never seem to agree with the actual computer program I wrote to do the work. Here is the scenario.

You have 3 buckets. each bucket contains 10 marbles.

In bucket A there are 2 white marbles and 8 black ones
In bucket B there are 3 white marbles and 7 black ones
In bucket C there are 4 white marbles and 6 black ones

You randomly select one marble from each bucket. What are the odds of selecting AT LEAST 2 white marbles?

Thanks in advance for any help ... want to learn how to do the require calculations, not just get the answer. Thanks again!
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August 27th, 2018, 10:05 AM   #2
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At least two means exactly 2 or exactly 3. Let w = number of whites.

$\text {P}(w = 2 \text { or } w = 3) = \text {P}(w = 2) + \text {P}(w = 3) - \text {P}(2 = w = 3).$

Now it is impossible for 2 = 3 $\implies \text {P}(2 = w = 3) = 0 \implies$

$\text {P}(w = 2 \text { or } w = 3) = \text {P}(w = 2) + \text {P}(w = 3) - 0 = \text {P}(w = 2) + \text {P}(w = 3).$

That is easy enough to figure out.

$a = \text { probability of choosing white from bucket A } = \dfrac{2}{10} = \dfrac{1}{5}.$

$x = \text { probability of choosing black from bucket A } = 1 - \dfrac{1}{5} = \dfrac{4}{5}.$

$b = \text { probability of choosing white from bucket B } = \dfrac{3}{10}.$

$y = \text { probability of choosing black from bucket B } = 1 - \dfrac{3}{10} = \dfrac{7}{10}.$

$c = \text { probability of choosing white from bucket C } = \dfrac{4}{10} = \dfrac{2}{5}.$

$z = \text { probability of choosing black from bucket C } = 1 - \dfrac{2}{5} = \dfrac{3}{5}.$

There is only one way to get exactly 3 whites, namely to pick 1 white from each bucket. Are those events independent? Yes.

So the probability of three whites $= abc = \dfrac{1 * 3 * 2}{5 * 10 * 5} = \dfrac{6}{250}.$

There are, however, six mutually exclusive ways to get exactly 2 whites. What are they? What are their probabilities?

Now add it all up, and you are done.

NOTE: This is not the most efficient way to compute the answer, but it is probably the most intuitive way for a beginner.

Last edited by JeffM1; August 27th, 2018 at 10:08 AM.
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