Bunuel wrote:
A certain board game has a row of squares numbered 1 to 100. If a game piece is placed on a random square and then moved 7 consecutive spaces in a random direction, what is the probability the piece ends no more than 7 spaces from the square numbered 49?
(A) 7%
(B) 8%
(C) 14%
(D) 15%
(E) 28%
If the piece is initially at square 49, then it doesn’t matter which direction it moves, it will still be no more than 7 spaces from square 49.
If the piece is initially at any one of the squares 50 to 63, inclusive, (a total of 14 squares), then it has to move to the left so that it ends no more than 7 spaces from square 49. Assuming there is an equal chance of moving to the right or left, the probability of moving to the left is 1/2.
If the piece is initially at any one of the squares 35 to 48, inclusive, (a total of 14 squares), then it has to move to the right so that it ends no more than 7 spaces from square 49. Assuming there is an equal chance of moving to the right or left, the probability of moving to the right is 1/2.
If the piece is initially at any one of the squares not mentioned above (i.e., squares 1 to 34 and squares 64 to 100), then there is no chance it can end no more than 7 spaces from square 49.
The final probability will be the weighted average of the probabilities of the initial square the piece is at. Therefore, the probability is:
1/100 x 1 + 14/100 x 1/2 + 14/100 x 1/2 + 71/100 x 0
1/100 + 14/100
15/100 = 15%
Answer: D