Question 1206109: 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
M = {the randomly chosen restaurant accepts master credit cards},
V
V = {the randomly chosen restaurant accepts visa credit cards}.
Which of the following shows independence between two events M
M and V
V? [CHECK ALL THAT APPLY]
A. P(V given M) is equal to P(V)
B. The events M and V are not disjoint
C. The events M and V are not disjoint
D.P(M given V) is equal to P(V)
e.P(M and V) is equal to P(M)×P(V)
F.P(M given V) is equal to P(M)
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52778)   (Show Source):
You can put this solution on YOUR website! .
This current post is a severely worsened formulation of the problem from another post
https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1206108.html
That different formulation was resolved in the referred post.
This given formulation in the current post is worsened so much that cannot be considered as a mathematical problem.
Answer by math_tutor2020(3816)  (Show Source):
You can put this solution on YOUR website!
It seems strange that answer choices B and C are identical. A typo perhaps?
M = restaurant accepts MasterCard
V = restaurant accepts Visa
Given information
P(V and M) = 0.37
P(M) = 0.50
P(V) = 0.60
We can then compute the following
P(V given M) = P(V and M)/P(M)
P(V given M) = 0.37/0.50
P(V given M) = 0.74
Note how this is not the same value as P(V) = 0.60
Therefore the equation P(V given M) = P(V) is false
Consequently it means events M and V are not independent. One event affects the other, or vice versa, or the two events are linked somehow.
Events M and V are not disjoint since P(V and M) = 0.37 is nonzero.
In other words, it's possible for both events to happen simultaneously. There is overlap between events.
Side note: An example of disjoint events would be "getting heads" and "getting tails" on the same coin on the same flip.
P(M given V) = P(M and V)/P(V)
P(M given V) = 0.37/0.60
P(M given V) = 0.61667 approximately
This is not the same as P(M) = 0.50, so the equation P(M given V) = P(M) is false. This is more proof that events M and V are not independent.
Because those events are not independent, P(M and V) = P(M)*P(V) is false.
Here is proof of such
P(M)*P(V) = 0.50*0.60 = 0.30 which doesn't match with P(M and V) = 0.37
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Summary:
If M and V were independent, then the following three equations would be true
P(V given M) = P(V)
P(M given V) = P(M)
P(M and V) = P(M)*P(V)
But we've shown that none of the equations are satisfied, so the events are not independent.
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