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In a large city, 37% of all restaurants accept both master and visa credit cards,
and 50% accept master cards and 60% accept visa cards. A tourist visiting the city
picks at random a restaurant at which to have lunch. Define the following events:
M= {the randomly chosen restaurant accepts master credit cards},
V= {the randomly chosen restaurant accepts visa credit cards}.
1) Are M and V disjoint?
2) Are M AND V independent?
3) Is M complement of V
4) What is the probability that a given restaurant does not accept master or visa credit cards?
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(1) The events (the sets) M and V have non-empty intersection of 37%.
Hence, they are NOT disjoint.
(2) To check if M and V indepentdent, we should check if P(M and V) is equal to P(M)*P(V).
We have P(M and V) = 0.37 (given). We also have P(M)*P(V) = 0.5*0.6 = 0.3.
Since 0.37 =/= 0.3, we conclude that events M and V are not independent.
(3) To check if M is complement of V, we should check TWO statements.
(a) whether M and V are disjoint ? - the answer is NO, see n.1 above
(b) whether P(M) + P(V) = 1 ? - Since 0.5 + 0.6 =/= 1, the answer is NO.
Having at least one NO answer of these two, (a) and (b), is just enought to say
that events M and V are not complementary.
(4) The probability that at least one restaurant accept M or V is
P(M or V) = P(M) + P(V) - P(M and V) = 0.5 + 0.6 - 0.37 = 0.73.
The probability that a given restaurant does not accept master or visa credit cards
is the complement to this value of 0.73.
So, the ANSWER to (4) is 1-0.73 = 0.27.
Solved.