Respuesta :
We assumed in this answer that the question b is, Are the events V and M independent?
Answer:
(a). The probability that a student has either a Visa card or a MasterCard is [tex] \\ P(V \cup M) = 0.88[/tex]. (b). The events V and M are not independent.
Step-by-step explanation:
The key factor to solve these questions is to know that:
[tex] \\ P(V \cup M) = P(V) + P(M) - P(V \cap M)[/tex]
We already know from the question the following probabilities:
[tex] \\ P(V) = 0.73[/tex]
[tex] \\ P(M) = 0.18[/tex]
The probability that a student has both cards is 0.03. It means that the events V AND M occur at the same time. So
[tex] \\ P(V \cap M) = 0.03[/tex]
The probability that a student has either a Visa card or a MasterCard
We can interpret this probability as [tex] \\ P(V \cup M)[/tex] or the sum of both events; that is, the probability that one event occurs OR the other.
Thus, having all this information, we can conclude that
[tex] \\ P(V \cup M) = P(V) + P(M) - P(V \cap M)[/tex]
[tex] \\ P(V \cup M) = 0.73 + 0.18 - 0.03[/tex]
[tex] \\ P(V \cup M) = 0.88[/tex]
Then, the probability that a student has either a Visa card or a MasterCard is [tex] \\ P(V \cup M) = 0.88[/tex].
Are the events V and M independent?
A way to solve this question is by using the concept of conditional probabilities.
In Probability, two events are independent when we conclude that
[tex] \\ P(A|B) = P(A)[/tex] [1]
The general formula for a conditional probability or the probability that event A given (or assuming) the event B is as follows:
[tex] \\ P(A|B) = \frac{P(A \cap B)}{P(B)}[/tex]
If we use the previous formula to find conditional probabilities of event M given event V or vice-versa, we can conclude that
[tex] \\ P(M|V) = \frac{P(M \cap V)}{P(V)}[/tex]
[tex] \\ P(M|V) = \frac{0.03}{0.73}[/tex]
[tex] \\ P(M|V) \approx 0.041[/tex]
If M were independent from V (according to [1]), we have
[tex] \\ P(M|V) = P(M) = 0.18[/tex]
Which is different from we obtained previously;
That is,
[tex] \\ P(M|V) \approx 0.041[/tex]
So, the events V and M are not independent.
We can conclude the same if we calculate the probability
[tex] \\ P(V|M)[/tex], as follows:
[tex] \\ P(V|M) = \frac{P(V \cap M)}{P(M)}[/tex]
[tex] \\ P(V|M) = \frac{0.03}{0.18}[/tex]
[tex] \\ P(V|M) = 0.1666.....\approx 0.17[/tex]
Which is different from
[tex] \\ P(V|M) = P(V) = 0.73[/tex]
In the case that both events were independent.
Notice that
[tex] \\ P(V|M)*P(M) = P(M|V)*P(V) = P(V \cap M) = P(M \cap V)[/tex]
[tex] \\ \frac{0.03}{0.18}*0.18 = \frac{0.03}{0.73}*0.73 = 0.03 = 0.03[/tex]
[tex] \\ 0.03 = 0.03 = 0.03 = 0.03[/tex]
