# Thread: Probability the First 2 chips are different colors

1. ## Probability the First 2 chips are different colors

Question: A bowl has three red and three blue chips. If two chips are randomly selected simultaneously, and then a third chip is selected, what is the probability that the first two chips were different colors, given that the third chip is red?

Attempt at Solution: R1 -first chip is red, B2 -second chip is blue, R3 -third chip is red
Using the conditional probability formula,
P(R1 and B2 | R3)
= P(R1 and B2 and R3) / P(R3)
= P(R1) P(B2 | R1) P(R3 | R1 and B2) / P(R3)
= (3/6) (3/5) (2/4) / P(R3)
n.b. already found P(R1 and B2 | R3) = P(B1 and R2 | R3)

n.b. c is used as the set complement notation
P(R3) = P(R1 and B2) P(R3 | R1 and B2)
. . . .. + P((R1 and B2)c) P(R3 | (R1 and B2)c)
P(R3) = (3/10) (2/4) + (1 - 3/10) ( ? )

I don't know how to find the probability that the third chip is red given that the first chip was red and the second chip was blue was NOT the case. If they are independent events, the probability would be the same as P(R3 | R1 and B2) = 1 - 3/10, but I don't know if they are necessarily independent events. What is the general formula for conditional probability with set complement?

Edit: Does P(R3 | (R1 and B2)c) = 13/21?

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