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Amber, Bella, and Cassey play a game that has 6 rounds. In each round there is one winner, and the outcomes of the rounds are independant. For each round the probability that Amber wins is 50%, and Bella is twice as likely to win as Cassey. What is the probability that Amber wins three rounds, Bella wins two rounds, and Cassey wins one round?

 Jan 18, 2019
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\(p_a + p_b+p_c = 1\\ \dfrac 1 2 + 2p_c + p_c = 1\\ 3p_c = \dfrac 1 2\\ p_c=\dfrac 1 6\\ p_b = \dfrac 1 3\)

 

\(P[a_3b_2c] = \dbinom{6}{3}\dbinom{3}{2} \left(\dfrac 1 2 \right)^3 \left(\dfrac 1 3 \right)^2\left(\dfrac 1 6 \right) = \dfrac{5}{36}\)

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 Jan 19, 2019

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