The probability that a student has a Visa card (event V) is .73. The probability that a student has a MasterCard (event M) is .18. The probability that a student has both cards is .03. (a) Find the probability that a student has either a Visa card or a MasterCard. (Round your answer to 2 decimal places.) Probability (b) In this problem, are V and M independent

We assumed in this answer that the question b is, Are the events V and M independent?

(a). The probability that a student has either a Visa card or a MasterCard is . (b). The events V and M are not independent.

Step-by-step explanation:

The key factor to solve these questions is to know that:

We already know from the question the following probabilities:

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

The probability that a student has either a Visa card or a MasterCard

We can interpret this probability as 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

Then, the probability that a student has either a Visa card or a MasterCard is .

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

[1]

The general formula for a conditional probability or the probability that event A given (or assuming) the event B is as follows:

If we use the previous formula to find conditional probabilities of event M given event V or vice-versa, we can conclude that

If M were independent from V (according to [1]), we have

Which is different from we obtained previously;

That is,

So, the events V and M are not independent.

We can conclude the same if we calculate the probability

, as follows:

Which is different from

In the case that both events were independent.

Notice that  

none of the above.

for one of the equations to be used in the process, there has to be a term in the equation since squared is .

since there is only a term in the first equation, we cannot derive a from completing the square.

solving with completing the square