In this exercise, you are asked to verify that the sum of the joint probabilities in a row or column of a joint probability distribution equals the marginal probability in that row or column. Consider the following joint probability distribution.

C1 ··· Cn P(Ri)

R1 P(R1 & C1) ··· P(R1 & Cn) P(R1)

· · ··· · ·

· · ··· · ·

· · ··· · ·

Rm P(Rm & C1) ··· P(Rm & Cn) P(Rm)

P(Cj) P(C1) ··· P(Cn) 1

a. Explain why R1 =

(R1 & C1) or ··· or (R1 & Cn)

.

b. Why are the events (R1 & C1),..., (R1 & Cn) mutually exclusive?

c. Explain why parts

(a) and

(b) imply that P(R1) = P(R1 & C1) +···+ P(R1 & Cn).

This equation shows that the first row of joint probabilities sums to the marginal probability at the end of that row. A similar argument applies to any other row or column.