A normal checkerboard has 8 rows and 8 columns of alternating colored squares (s

Question

A normal checkerboard has 8 rows and 8 columns of alternating colored squares (see picture of antique checkerboard).

Suppose that you instead have a 6 by 6 checkerboard with alternating brown and white squares. You randomly place 4 pieces on the checkerboard. For each of the following, give your answers correct to six decimal places.

(a) What is the probability that all of the pieces will be in column #1?


(b) What is the probability all of the pieces will be in the same column (all in column #1 or all in column #2 or all in column #3, etc.)?


(c) What is the probability that all of the pieces will be placed on white squares?


(d) What is the probability that all of the pieces will be placed on squares of the same color (all on white or all on brown)?

Explanation / Answer

a) the number of ways to place 4 pieces 1st column = 6C4

total sample space = 36C 4

therefore, probability that all pieces will be in column 1 = 6C4 / 36C4

= 15 / 58905 = .0002546

b) probability that all pieces will be in same column = 6*.0002546 = 0.001528

c) number of ways to put 4 pieces in 18 white = 18C4 = 3060

probability = 3060/ 58905 = 0.01595

d)  probability that all of the pieces will be placed on squares of the same color = 2* 0.01595 = 0.1039

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