**VJesus12 wrote:**In order to make the national tennis team, Matt has to play a three-game series against Larry and Steve, and in doing so win two games in a row. He's given a choice, however: he can choose the order in which he plays against his opponents but cannot play the same opponent in consecutive games (so he could play Larry-Steve-Larry OR Steve-Larry-Steve). Assuming that Matt chooses the three-game sequence that maximizes his probability of making the national team, is his probability of making the team greater than 51%?

(1) Matt's probability of beating Steve is better than Matt's probability of beating Larry

(2) The probability that Matt beats Larry is 30%.

The OA is the option B.

I am really confused on this questions. Experts, can you help me here, please? I don't know how to solve this DS question.

We are given the following information.

- Matt has to play against Larry and against Steve.

- Matt has to win two games in a row. So, there can be one of the three sequences: Win-Win-Win, Win-Win-Lose or Lose-Win-Win

- Matt can choose the order in which he plays against his opponents but cannot play the same opponent in consecutive games (so he could play

Larry-Steve-Larry OR

Steve-Larry-Steve)

- Matt chooses the three-game sequence that maximizes his probability of making the national team. If Matt's probability of winning against Larry is greater than that against Steve, he has the right to choose Larry-Steve-Larry sequence; however, if Matt's probability of winning against Steve is greater than that against Larry, he has the right to choose Steve-Larry-Steve sequence.

We have to determine if Matt's probability of making the team greater than 51%.

Let's take each statement one by one.

(1) Matt's probability of beating Steve is better than Matt's probability of beating Larry

Since Matt's probability of beating Steve is better than Matt's probability of beating Larry, he would obviously choose Steve-Larry-Steve sequence.

Thus, the probability of making the national team in a three-game sequence = p(S)*p(L)*p(S); where p(S) = probability of winning against Steve and p(L) = probability of winning against Larry

Case 1: Say p(S) = 1, p(L) = 4/5, then p(S)*p(L)*p(S) = 1*4/5*1 = 80% > 51%. Matt makes it. The answer is Yes.

Case 2: Say p(S) = 1/2, p(L) = 1/3, then p(S)*p(L)*p(S) = 1/2*1/3*1/2 = 1/12 = 16.67% < 51%. Matt does not make it. The answer is No.

No unique answer. Insufficient.

(2) The probability that Matt beats Larry is 30%.

Since Matt's probability of winning against Larry is 30% (< 51%), if his probability of winning against Steve is also less than 51%, he cannot make it. The answer is No.

Let's take the best scenario. If his probability for the 3-game sequence is more than 51%, the answer would then be Yes. Thus, the case of No Unique Answer; however, if his probability for the 3-game sequence is STILL LESS than 51%, the answer would then be No. Thus, the case of A Unique Answer.

BEST SCENARIO:

Say the probability of winning against Steve = 1, then Matt would choose the Steve-Larry-Steve sequence.

p(S)*p(L)*p(S) = 1*3/10*1 = 3/10 = 30% < 51%. Even in the best scenario, his probability of winning is STILL LESS than 51%. The answer is No, Matt cannot make it to the national team. Sufficient.

The correct answer:

B
Hope this helps!

-Jay

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