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GMAT/Data Sufficiency - Funball Match
Funball Match
In the game of Funball, each batter can either hit a home run, hit a single, or strikeout, and the likelihood of each outcome is completely determined by the opposing pitcher. A Funball batter scores a point for their team by advancing sequentially through each of four "bases", according to the following rules:
Home run: The batter and any players already on a base advance through all four bases.
Single: The batter advances to first base, and any players already on a base advance one base each.
Strikeout: No one advances any bases, and the batter loses his/her turn.
If the batting team has a runner on first base, which pitcher (Roger or Greg) is more likely to allow a point before recording a strikeout?
(1) Greg is twice as likely as Roger to allow a single, and four times as likely as Roger to record a strikeout.
(2) Greg is twice as likely as Roger to allow a single, and one fourth as likely as Roger to allow a home run.
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.
| ОТВЕТЫ И КОММЕНТАРИИ |
| Решение |
If Roger pitches, let  be the probability of a home run,  be the probability of a single, and  be the probability of a strikeout (all batters face these same probabilities, since the problem states that these probabilities are completely determined by the pitcher).
If Greg pitches, let  be the probability of a home run,  be the probability of a single, and  be the probability of a strikeout.
The following are the only three event sequences in which no points will score before a strikeout occurs:
1. The current batter strikes out. (K)
2. The current batter hits a single, and the next batter strikes out. (SK)
3. The current batter hits a single, the next batter hits a single, and the following batter strikes out. (SSK)
(Note that if three consecutive batters hit singles, or if any batter hits a home run, then the batting team will score at least one point.)
If Roger pitches, the probability of any one of the three sequences mentioned above occurring is:
If Greg pitches, the probability of any one of the three sequences occurring is:
We need to be able to determine whether R or G is greater in order to solve the problem.
Statement (1) gives us the following:
 (Note: We also know that  ,  ,  , and  cannot be equal to 0).
We can substitute these equations in the probability expressions for G:
Since all of the unknowns are positive values, we can see from these equations that G will always be greater than R. This means that Greg is more likely than Roger to record a strikeout before allowing a point (which, in turn, means that Rorger is more likely than Greg to allow a point before recording a strikeout.) Therefore statement (1) is sufficient to solve the problem.
Statement (2) gives us the following:
 and  (Note: We also know that that  ,  ,  , and  cannot be equal to 0).
Note that the probabilities G and R are expressed in terms of that  ,  ,  , and  . Whereas statement (1) tells us that and  and  (and therefore that G > R, solving the problem), statement (2) lacks any information about the size of  relative to  . (Information about the size of  relative to  does not help us at all since neither of these variables are part of the probability expressions for G and R.)
As such, the information in statement (2) is insufficient to solve the problem.
Therefore, the correct answer to this problem is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
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School Closing
There is a 10% chance that it won't snow all winter long. There is a 20% chance
that schools will not be closed all winter long. What is the greatest possible
probability that it will snow and schools will be closed during the winter?
(A) 55%
(B) 60%
(C) 70%
(D) 72%
(E) 80%
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