Question

Of the 10,000 students at a certain university, 7000 have Visa cards, 6000 have MasterCards, and 5000 have both. Suppose that a student is randomly selected. a. What is the probability that the selected student has a Visa card? b. What is the probability that the selected student has both cards? c. Suppose you learn that the selected individual has a Visa card (was one of the 7000 with such a card). Now what is the probability that this student has both cards? d. Are the events has \(a\) Visa card and has a MasterCard independent? Explain. e. Answer the question posed in Part (d) if only 4200 of the students have both cards.

Short answer

The probabilities are as follows - a) The probability that a student has a Visa card is 0.7. b) The probability that a student has both cards is 0.5. c) Given that a student has a Visa card, the probability that they also have a MasterCard is 0.714. d) The events 'has a Visa card' and 'has a MasterCard' are not independent. e) Even with the information that 4200 students have both cards, the events are not independent.

Step-by-step solution

Calculate the Probability of Having a Visa Card

The total number of students is 10,000 and the number of students having Visa cards is 7000. So, the probability that a randomly selected student has a Visa card is given by dividing the number of students having a Visa card by the total number of students. So, it's \( \frac{7000}{10000} = 0.7. \)

Calculate the Probability of Having Both Cards

The number of students having both Visa and MasterCard is 5000. Thus, the probability that a randomly selected student has both cards can be found by dividing the number of students having both cards by the total number of students. So, it's \( \frac{5000}{10000} = 0.5. \)

Find the Conditional Probability

We know that the student selected has a Visa card. Now we need to find the probability that this student also has a MasterCard. Out of 7000 Visa cardholders, 5000 have both cards. Therefore, the conditional probability is \( \frac{5000}{7000} = 0.714. \)

Check if the Events are Independent

Two events A and B are independent if P(A|B) = P(A), which means the probability of event A given event B is the same as the probability of event A. Here, it is given that P(Visa card) = 0.7 and P(Both cards | Visa card) = 0.714. Since these probabilities are not equal, the two events are not independent.

Repeat the Independence Test with New Data

Now, with the new information that only 4200 students have both cards, we have to check if the events 'has a Visa card' and 'has a MasterCard' are independent or not. We repeat the previous calculation of conditional probability and find that P(Both cards | Visa card) = \( \frac{4200}{7000} = 0.6. \) Since this is not equal to P(Visa card), the two events are still not independent.

Key concepts

Conditional Probability

Conditional probability is a measure of the likelihood of an event occurring given that another event has already occurred. In essence, it adjusts the probability of an event based on the new information that another event has taken place. For example, in our textbook exercise, we initially know the probability of a student having a Visa card is 0.7. However, once we know the student has a Visa card, we are asked to find the probability that they also have a MasterCard. This is known as conditional probability because it is the probability of the second event (having a MasterCard), given that the first event (having a Visa card) has occurred.

The formula used to calculate conditional probability is: \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), where \( P(A|B) \) is the probability of event A given event B, \( P(A \cap B) \) is the probability of both events A and B occurring, and \( P(B) \) is the probability of event B. In the solution steps, we saw that out of the 7000 Visa cardholders, 5000 also had a MasterCard, resulting in a conditional probability of \( \frac{5000}{7000} = 0.714 \), answering part (c) of the exercise.

Independent Events

Independent events in probability are two or more events where the occurrence of one does not affect the occurrence of another. For events to be independent, the probability of one event occurring must remain constant regardless of whether the other event occurs.

As illustrated in the solution, we check for independence by comparing the unconditional probability of having a Visa card with the conditional probability of having both a Visa and a MasterCard given the student already has a Visa card. Independence would imply that these probabilities are the same. However, in our exercise, the fact that the two probabilities \( (P(Visa card) = 0.7 \) and \( P(Both cards | Visa card) = 0.714) \) differ indicates that the events are dependent. Likewise, even when the data changes to 4200 students having both cards, the revised conditional probability of \( 0.6 \) still differs from the probability of having a Visa card alone, reinforcing that these are dependent events.

Probability Theory

Probability theory is a branch of mathematics that deals with quantifying the likelihood of different outcomes. It allows us to make informed statements about uncertain events and to model and analyze randomness and chance. This theory includes various concepts such as outcomes, events, independence, and conditional probabilities.

It's vital in statistics and is used to draw conclusions and make predictions based on data. In our exercise, we used probability theory to perform calculations such as the probability of randomly selecting a student with a Visa or MasterCard and assessing the independence of two events. By systematically applying these concepts, we are able to extract meaningful information from the given data, which is the essence of statistical analysis using probability theory.