Using (8 x 8) checkerboard. A checker is in the bottom square of the fifth column. It will randomly move diagonally left or right one space forward. What is the probability the checker ends up in the top square of the second column?
I got 19.4%. Is that right?
To find the probability that the checker ends up in the top square of the second column, we can analyze the possible paths it can take.
Since the checker is initially in the bottom square of the fifth column, it has two possible moves: diagonally left or diagonally right. Let's denote these moves as "L" for left and "R" for right.
In order to reach the top square of the second column, the checker must make exactly 6 moves and have 2 "L" moves and 4 "R" moves (in any order).
Now, let's calculate the number of possible paths the checker can take to end up in the top square of the second column.
First, we need to determine the number of ways to order the 6 moves (2 "L"s and 4 "R"s). This can be calculated using the concept of permutations.
The number of permutations of a set of n elements, where there are k elements of one type and m elements of another type, is given by:
P = n! / (k! * m!)
In this case, n = 6, k = 2 (number of "L" moves), and m = 4 (number of "R" moves).
P = 6! / (2! * 4!) = (6 * 5 * 4 * 3 * 2 * 1) / ((2 * 1) * (4 * 3 * 2 * 1)) = 15
So, there are 15 possible paths the checker can take to reach the top square of the second column.
Next, let's calculate the total number of possible paths that the checker can take in general.
Starting from the bottom square of the fifth column, the checker has two possible moves at each step. Since there are 6 steps in total (6 additional moves after the initial position), the total number of possible paths can be calculated using the concept of combinations.
The number of combinations for selecting k objects from a set of n objects is given by:
C = n! / (k! * (n - k)!)
In this case, n = 6 (number of additional moves after the initial position) and k = 6 (total number of steps).
C = 6! / (6! * (6 - 6)!) = 1
So, there is only 1 path in total that the checker can take.
Finally, let's calculate the probability that the checker ends up in the top square of the second column.
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
Number of favorable outcomes = 15 (as calculated earlier)
Total number of possible outcomes = 1 (as calculated earlier)
Probability = 15 / 1 = 15
Therefore, the probability that the checker ends up in the top square of the second column is 15, not 19.4%. Please note that probabilities are typically expressed as decimal values between 0 and 1, or as percentages between 0% and 100%.