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  

answer: 1) twenty-seven times y

              2) twenty-seven multiplied by y

by definition, an arc subtended by a central angle of some given measure has the same measure. so .