25 points 1. The WTA is trying to organize a tennis tournament with 5 of the top

Question

25 points 1. The WTA is trying to organize a tennis tournament with 5 of the top 10 players. Assume Maria Sharapova, Caroline Wozniacki, Venus Williams, and Serena Williams are in the top 10. Venus and Serena will play only if both are invited. Otherwise, both don't play Wozniacki will not play if Sharapova plays. Sharapova and the other players have no restrictions. What is the probability that Sharapova and the Williams sisters play in this tournament? 20 points 2. The probability that a missile can sink a warship is 0.3; the probability that the missile can only make some damage is 0.5; and the probability that the missile misses the warship is 0.2. Further, the warship will sink if 2 damage shots hit. What is the probability that at most 4 missiles are needed to sink the warship?

Explanation / Answer

1)

There are total 10 player. 5 of them can be selected in 10C5 ways

If Sharapova and Williams sister play, we only need to have other 2 from 7 players. As Sharapova is playing, Caroline will not play. This means there are only 6 players available for selection of 2. This can be done in 6C2 ways.

Required probability = 6C2/10C5 = 15/252 = 0.0595

2)

P(S) = 0.3, P(D) = 0.5 and P(M) = 0.2

At most 4 missiles are required to sink the warship, probability = P(S) + P(M)*P(S) + P(M)*P(M)*P(S) + P(M)*P(M)*P(M)*P(S) + P(D)*P(S) + P(D)*P(M)*P(S) + P(D)*P(M)*P*(M)*P(S) + P(D)*P(D) + P(D)*P(M)*P(D) + P(D)*P(M)*P(M)*P(D) + P(M)*P(D)*P(M)*P(D) + P(M)*P(M)*P(D)*P(D)

probability = ( 0.3 + 0.2*0.3 + 0.2*0.2*0.3 + 0.2*0.2*0.2*0.3 ) + (0.5*0.3 + 0.5*0.2*0.3 + 0.5*0.2*0.2*0.3 ) + (0.5*0.5 + 0.5*0.2*0.5 + 0.5*0.2*0.2*0.5 * 0.5*0.2*0.5*0.2 +  0.2*0.2*0.5*0.5)

probability = 0.8705

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