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Chapter 4: Problem 41

Briefly explain the difference between the marginal and conditional probabilities of events. Give one example of each.

Short Answer

Expert verified
Marginal probability is the likelihood of an event happening on its own, while conditional probability is the likelihood of an event happening given that another event has occurred. For example, the marginal probability of drawing a red card from a deck is 0.5. However, the conditional probability of drawing a red card given that the card is an Ace is also 0.5.

Step by step solution

01

Understanding Marginal Probability

The marginal probability is the probability of an event occurring. It doesn't take into account any information about any other event. To illustrate, consider a simple event, such as the probability of drawing a red card from a standard deck of cards. The deck contains 52 cards, of which 26 are red. Therefore, the marginal probability of drawing a red card is \( P(Red) = \frac{26}{52} = 0.5 \) .
02

Understanding Conditional Probability

Conditional probability is the probability of an event occurring, given that another event has already occurred. If you have two events \(A\) and \(B\), the conditional probability of \(A\) happening, given that \(B\) has happened, is usually written as \(P(A|B)\). For example, consider the scenario of drawing a red card from a standard deck of cards, given that the drawn card is an Ace. There are 4 aces in the deck and 2 of them are red. So, the conditional probability of drawing a red card given that it's an ace is \( P(Red|Ace) = \frac{2}{4} = 0.5 \) .
03

Distinguishing Between Marginal and Conditional Probability

To summarize, marginal probability looks at the likelihood of a single event happening irrespective of any other events. Conditional probability, on the other hand, takes into account the occurrence of some other event. In our examples, the marginal probability represented the likelihood of drawing a red card from the deck, regardless of its rank, while the conditional probability represented the likelihood of drawing a red card knowing that the card drawn is an ace.

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Most popular questions from this chapter

In a group of 10 persons, 4 have a type A personality and 6 have a type B personality. If two persons are selected at random from this group, what is the probability that the first of them has a type A personality and the second has a type B personality? Draw a tree diagram for this problem

Explain the meaning of the union of two events. Give one example.

A random sample of 400 college students was asked if college athletes should be paid. The following table gives a two-way classification of the responses. $$\begin{array}{lcc} \hline & \text { Should Be Paid } & \text { Should Not Be Paid } \\ \hline \text { Student athlete } & 90 & 10 \\ \text { Student nonathlete } & 210 & 90 \end{array}$$ a. If one student is randomly selected from these 400 students, find the probability that this student i. is in favor of paying college athletes ii. favors paying college athletes given that the student selected is a nonathlete iii. is an athlete and favors paying student athletes iv. is a nonathlete or is against paying student athletes b. Are the events "student athlete" and "should be paid" independent? Are they mutually exclusive? Explain why or why not.

Consider the following addition rule to find the probability of the union of two events \(A\) and \(B\) : $$P(A \text { or } B)=P(A)+P(B)-P(A \text { and } B)$$ When and why is the term \(P(A\) and \(B)\) subtracted from the sum of \(P(A)\) and \(P(B)\) ? Give one example where you might use this formula.

The probability that a randomly selected college student attended at least one major league baseball game last year is .12. What is the complementary event? What is the probability of this complementary event?
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