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In the game of Twister, a spinner randomly selects either an arm or a leg, and also selects one of four colors, one of which is red, each with equal probability, and players have to move the appropriate body part to the appropriately colored spot on the ground. There are four players. Each player spins once, and makes the move the spinner selects. What is the probability that in these four spins, there will be exactly two moves to a red spot, and the body part selected to move will be an arm exactly $3$ times?

 Aug 1, 2021
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There are C(4,2) = 6 ways to choose the two moves to the red spot.  The probability of moving to a red spot is 1/4, and since we have two moves the probability is 1/16.

 

There are C(4,3) = 4 ways to choose the three moves with an arm.  Each arm happens with probability 1/2, so the total is 1/8.

 

The cases are independent, so we multiply them to get 6*1/16 + 4*1/8 = 3/16.

 Aug 1, 2021

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