Two fair die are thrown. M is the event described by "The sum of the scores is 10" and N is the event described by "The difference between the scores is 3"....

Answer Details

Two fair dice are thrown, giving 36 equally likely ordered outcomes.

(a) Elements.

\(M\) = "sum of the scores is 10":

\[M = \{(4,6),\ (5,5),\ (6,4)\}.\]

\(N\) = "difference between the scores is 3":

\[N = \{(1,4),\ (4,1),\ (2,5),\ (5,2),\ (3,6),\ (6,3)\}.\]

So \(n(M) = 3\) and \(n(N) = 6\).

(b) P(M or N). Checking for common outcomes: the pairs in \(M\) have differences 2, 0, 2, so \(M \cap N = \varnothing\). Hence

\[P(M \cup N) = P(M) + P(N) = \frac{3}{36} + \frac{6}{36} = \frac{9}{36} = \frac{1}{4}.\]

(c) Yes, \(M\) and \(N\) are mutually exclusive, because they have no outcome in common (\(M \cap N = \varnothing\)); no single throw can give a sum of 10 and a difference of 3 at the same time.