A certain board game has a row of squares numbered 1 to 100.

Yes, "a game piece is placed on a random square and then moved 7 consecutive spaces in a random direction" just means that a game piece is placed on a random square.

There are three sections of interest in this problem
1) locations (42-48) and user has to move the piece to it's right -> probability P1 -> (7/100) * (1/2)
2) location 49 is selected (User can move it any direction and is still in the limit) -> probability P2 -> 1/100
3) locations (50-56) and user has to move the piece to it's left -> probability P3 -> (7/100) *(1/2)

So total probability is (P1+P2+P3) -> 8/100 i.e 8%

Let me know your thoughts

7 spaces from 49 to the right ---> 56
7 spaces from 49 to the left ----> 42

56-42 = 14 +1 = 15

therefore 15/100 = 15% (D)

I am getting 28 as my answer... Can somebody explain why are we ignoring extended ranges?

Explanation - I can select 35, move right, end up in 42 (49- 7) or else select 63, move left, end up in 56 (49+7). So why not 28 (63-38)

Posted from GMAT ToolKit

First of all you should include 35 and 63. So, the range is 30 (from 28 to 62, inclusive). Next, sine the game piece is moved in a random direction, then in half of the case it will move in the wrong direction (away from the range 42-56, inclusive), thus the probability is 15/100.

Hope it's clear.

Ok ... May be I am not getting it yet and Confusing extra numbers 35, 30, 28, 62 above ..

However if I concentrate on last part of sentence and if I understand it correct, it means that there are two ways I can fall at a number (left or right). But from either way there are equal chances of falling into the range and also equal chances of going out of the range (i.e. 42-56). Since these chances cancel each other out, it is not required to consider the direction to reach a number.. Just the range of number maters .. Am I right?

Posted from GMAT ToolKit

Yes, we should simply consider the range from 42 to 56, inclusive.

Numbered from 1 to 100
Piece can move to either 7 units right or left and should be within 7 spaces from 49
--> Final position of piece should be from 35 to 48 moving only right
Or
From 50 to 63 moving only left
Or
At 49 exactly.

So, the favorable range of values = 63 - 35 = 28/2 + 1 = 15

Probability = 15/100

IMO Option D

Pls Hit Kudos if you like the solution

I don't think the question setup makes sense, because it's not clear what happens in the game if you place the piece on square #2, say, and then want to randomly move it left 7 spaces. It would make sense if the board was circular, so things 'wrapped around', but then the piece's location will be purely random after we move it, so we don't care how we're moving the tile. We'd just need to count how many squares are no more than 7 spaces from square #49. There are 15 of those, so the probability is 15/100.

If you did want to account for how the tile moves, you can divide the problem into cases (assuming the game board wraps around, so any starting location and any direction of movement is legal) :

If the piece starts on square #49, it will automatically be no more than 7 spaces from that square after we move it. The probability this happens is 1/100.

If the piece starts on any square from #35 to #48, then half the time (when we move it to the right) it will end up no more than 7 spaces from square #49. The probability this happens is (1/2)(14/100) = 7/100

Similarly if the piece starts on any square from #50 to #63, then half the time (when we move it to the left) it will end up no more than 7 spaces from square #49. The probability this happens is (1/2)(14/00) = 7/100

Adding the three cases above, we get the answer, 15/100.

Case 1- When piece will move only forward
Probability to move forward= 1/2
Number of squares that piece can be so that it will not end up 7 spaces from the square numbered 49= (49-35)+1=15
Probability= (15/100)*(1/2)

Case 2- When piece will move only backwards
Probability to move backwards= 1/2
Number of squares that piece can be so that it will not end up 7 spaces from the square numbered 49= (63-49)+1=15
Probability= (15/100)*(1/2)

Total probability= (15/100)*(1/2)+(15/100)*(1/2)=15/100

3 scenarios.

1) we place the piece at 49. The probability of that happening is 1/100

2) If we chose to move only left after placing the piece, then we get 14 squares (50 to 63). Probability of choosing 14 out of 100 is 14/100.

3) If we chose to move only right after placing the piece, then we get 14 squares (35 to 48). Probability of choosing 14 out of 100 is 14/100.

Add all 3 probabilities. Option E is the answer.

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

Solution:

The total number of squares = 100 is the sample space here.

Let the game piece be at the 49th square.

A movement of 7 consecutive spaces occurs

=>Maximum value of the square where the game piece can land = 49 + 7 =56

and

Minimum value of the square where the game piece can land = 49 - 7 = 42

Total number of squares that thus can be covered with this movement = 56 - 42 + 1 = 15

=>Probability = 15/100 =0.15 (option d)

Devmitra Sen
GMAT SME