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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

Two thousand randomly selected adults were asked whether or not they have ever shopped on the Internet. The following table gives a two-way classification of the responses. $$\begin{array}{lcc} \hline & \text { Have Shopped } & \text { Have Never Shopped } \\ \hline \text { Male } & 500 & 700 \\ \text { Female } & 300 & 500 \end{array}4$ a. If one adult is selected at random from these 2000 adults, find the probability that this adult i. has never shopped on the Internet ii. is a male iii. has shopped on the Internet given that this adult is a female iv. is a male given that this adult has never shopped on the Internet b. Are the events "male" and "female" mutually exclusive? What about the events "have shopped" and "male?" Why or why not? c. Are the events "female" and "have shopped" independent? Why or why not?

A random sample of 250 juniors majoring in psychology or communication at a large university is selected. These students are asked whether or not they are happy with their majors. The following table gives the results of the survey. Assume that none of these 250 students is majoring in both areas. $$ \begin{array}{lcc} \hline & \text { Happy } & \text { Unhappy } \\ \hline \text { Psychology } & 80 & 20 \\ \text { Communication } & 115 & 35 \end{array} $$ a. If one student is selected at random from this group, find the probability that this student is i. happy with the choice of major ii. a psychology major iii. a communication major given that the student is happy with the choice of majon iv. unhappy with the choice of major given that the student is a psychology major v. a psychology major and is happy with that major vi. a communication major \(o r\) is unhappy with his or her major b. Are the events "psychology major" and "happy with major" independent? Are they mutually exclusive? Explain why or why not.

Define the following two events for two tosses of a coin: \(A=\) at least one head is obtained \(B=\) both tails are obtained a. Are \(A\) and \(B\) mutually exclusive events? Are they independent? Explain why or why not. b. Are \(A\) and \(B\) complementary events? If yes, first calculate the probability of \(B\) and then calculate the probability of \(A\) using the complementary event rule.

A random sample of 2000 adults showed that 1320 of them have shopped at least once on the Internet. What is the (approximate) probability that a randomly selected adult has shopped on the Internet?

A random sample of 250 adults was taken, and they were asked whether they prefer watching sports or opera on television. The following table gives the two-way classification of these adults $$\begin{array}{lcc} \hline & \begin{array}{c} \text { Prefer Watching } \\ \text { Sports } \end{array} & \begin{array}{c} \text { Prefer Watching } \\ \text { Opera } \end{array} \\ \hline \text { Male } & 96 & 24 \\ \text { Female } & 45 & 85 \end{array}$$ a. If one adult is selected at random from this group, find the probability that this adult i. prefers watching opera ii. is a male iii. prefers watching sports given that the adult is a female iv. is a male given that he prefers watching sports \(\mathbf{v}\). is a female and prefers watching opera vi. prefers watching sports or is a male b. Are the events "female" and "prefers watching sports" independent? Are they mutually exclusive? Explain why or why not.
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