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"....
Question 1 Report
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".
(a) Write out the elements of M and N.
(b) Find the probability of M or N.
(c) Are M and N mutually exclusive? Give reasons.
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.
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