manik11 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 are better than Matt's probability of beating Larry
(2) The probability that Matt beats Larry is 30%
OA : B
Source : Veritas Prep
For statement 1, you can try some extreme scenarios. Say Matt's probability of beating Steve is 100% and his probability of beating Larry is 99%. No need to do any math here- clearly his probability of winning two in a row is over 51%. So that's a YES. Now say Matt's probability of beating Steve is 1% and his probability of beating Larry is .5%. Clearly, his probability of winning two in a row is not over 51%, so that's a NO. Statement 1 is not sufficient.
Statement 2 is more complex than it appears at first glance. First, if Matt has to alternate between Larry and Steve, he'll need to beat both players in order to win two in a row. We know that Matt has a 30% probability of beating Larry. (And a 70% chance of losing to Larry.) Obviously the highest possible probability that Matt will beat Steve is 100%. So let's examine that scenario.
Now he has a choice. He can play Larry-Steve-Larry or he can play Steve-Larry-Steve. It may feel counterintuitive, but Matt's odds of winning two straight are better with the first scenario. (Think of it this way: he has to beat Larry in order to win two in a row. In the first scenario, he gets two cracks at Larry. In the second one, he only gets one.)
So then what is the probability that Matt wins at least two in a row, if he plays Larry-Steve-Larry? Well, we know he beats Steve, if there's a 100% probability of victory. So as long as Matt beats Larry at least once, he'll win two in a row. (He could win the first two or the last two or all three.)
P(beat Larry at least once) = 1 - P(never beat Larry)
P(never beat Larry) = .7 *.7 = .49
1 - P(never beat Larry) = 1 - .49 = .51. Meaning the best possible scenario entails a 51% chance that Matt wins at least two in a row. Therefore, we know that the probability is never higher than 51%, and thus statement 2 alone is sufficient. Tricky question.
