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The question is this:

"An urn is filled with 8 green balls, 2 red balls, and 6 orange balls. Three balls are selected without replacement."

What is the probability that exactly one ball is orange?

I know I could just use the binomial formula if each event were independent (i.e. three balls were selected [replacement). But I'm not sure how to find the probability in this case because they are dependent events, and the order in which the orange ball is picked affects the probability. I can see three different scenarios of one orange ball being picked:with

P(orange picked)*P(not orange picked)*P(not orange picked)

or

P(not orange picked)*P(orange picked)*P(not orange picked)

or

P(not orange picked)*P(not orange picked)*P(orange picked)

The first case corresponds to:

(6/16)(10/15)(9/14)= 9/56

The second case corresponds to:

(10/16)(6/15)(9/14)= 9/56

The third case corresponds to:

(10/16)(9/15)(6/14)= 9/56

Would the probability that exactly one ball is orange just be 3(9/56), or 27/56?

Would a similar process be done if I wanted to know the probability that at most one ball is red?

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# Probability of exactly one when events are dependent

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