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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

A player plays a roulette game in a casino by betting on a single number each time. Because the wheel has 38 numbers, the probability that the player will win in a single play is \(1 / 38\). Note that each play of the game is independent of all previous plays a. Find the probability that the player will win for the first time on the 10 th play. b. Find the probability that it takes the player more than 50 plays to win for the first time c. The gambler claims that because he has 1 chance in 38 of winning each time he plays, he is certain to win at least once if he plays 38 times. Does this sound reasonable to you? Find the probability that he will win at least once in 38 plays

Given that \(A\) and \(B\) are two independent events, find their joint probability for the following. a. \(P(A)=.61\) and \(P(B)=.27\) b. \(P(A)=.39\) and \(P(B)=.63\)

A consumer agency randomly selected 1700 flights for two major airlines, \(\mathrm{A}\) and \(\mathrm{B}\). The following table gives the two-way classification of these flights based on airline and arrival time. Note that "less than 30 minutes late" includes flights that arrived early or on time. $$\begin{array}{lccc} \hline & \begin{array}{c} \text { Less Than 30 } \\ \text { Minutes Late } \end{array} & \begin{array}{c} \text { 30 Minutes to } \\ \text { 1 Hour Late } \end{array} & \begin{array}{c} \text { More Than } \\ \text { 1 Hour Late } \end{array} \\ \hline \text { Airline A } & 429 & 390 & 92 \\ \text { Airline B } & 393 & 316 & 80 \\ \hline \end{array}$$ a. If one flight is selected at random from these 1700 flights, find the probability that this flight is \(\mathrm{i}\), more than 1 hour late ii. less than 30 minutes late iii. a flight on airline A given that it is 30 minutes to 1 hour late iv. more than 1 hour late given that it is a flight on airline \(\mathrm{B}\) b. Are the events "airline A" and "more than 1 hour late" mutually exclusive? What about the events "less than 30 minutes late" and "more than 1 hour late?" Why or why not? c. Are the events "airline \(\mathrm{B}\) " and " 30 minutes to 1 hour late" independent? Why or why not?

Given that \(A\) and \(B\) are two mutually exclusive events, find \(P(A\) or \(B\) ) for the following. a. \(P(A)=.25\) and \(P(B)=.27\) b. \(P(A)=.58\) and \(P(B)=.09\)

According to a survey of 2000 home owners, 800 of them own homes with three bedrooms, and 600 of them own homes with four bedrooms. If one home owner is selected at random from these 2000 home owners, find the probability that this home owner owns a house that has three or four bedrooms. Explain why this probability is not equal to \(1.0 .\)
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