Soren has two jobs. During the day he works as an office clerk, and in the evening he works as a cashier. His office job pays hi
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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<em> </em>. (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, <em>the probability that a student has either a Visa card </em><em>or</em><em> a MasterCard is </em>.<em> </em>
Are the events V and M independent?
A way to solve this question is by using the concept of <em>conditional probabilities</em>.
In Probability, two events are <em>independent</em> when we conclude that
[1]
The general formula for a <em>conditional probability</em> 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 <em>were independent</em>.
Notice that