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Chapter 7: Problem 63

Each day, there is a \(40 \%\) chance that you will sell an automobile. You know that \(30 \%\) of all the automobiles you sell are two-door models, and the rest are four-door models.

Short Answer

Expert verified
The probability of selling a two-door automobile each day is \(0.12\) or \(12\%\) and the probability of selling a four-door automobile each day is \(0.28\) or \(28\%\).

Step by step solution

01

Identify the given probabilities

We are given the following probabilities: - P(Selling an automobile) = 40% = 0.4 - P(Two-door | Sold) = 30% = 0.3 - P(Four-door | Sold) = 100% - 30% = 70% = 0.7
02

Calculate the probability of selling a two-door and a four-door automobile each day

We need to find the probabilities P(Two-door) and P(Four-door). We can use the conditional probability formula for this purpose: P(A | B) = P(A and B) / P(B) So, for selling a two-door automobile: P(Two-door | Sold) = P(Two-door and Sold) / P(Sold) P(Two-door and Sold) = P(Two-door | Sold) * P(Sold) = 0.3 * 0.4 = 0.12 And for selling a four-door automobile: P(Four-door | Sold) = P(Four-door and Sold) / P(Sold) P(Four-door and Sold) = P(Four-door | Sold) * P(Sold) = 0.7 * 0.4 = 0.28
03

Presenting the results

The probability of selling a two-door automobile each day is 0.12 or 12%. The probability of selling a four-door automobile each day is 0.28 or 28%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability
Understanding conditional probability is essential for interpreting how likely it is that something will happen given that another event has occurred. Take the example of selling automobiles, with a daily chance of a sale being 40%. Let's delve deeper into the case where we want to know the likelihood of a two-door model being sold, given that a car sale has occurred. This is a classic scenario for applying conditional probability.

By definition, the conditional probability of event A, given event B, is expressed as: \[ P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \].

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Most popular questions from this chapter

You are given a transition matrix \(P\) and initial distribution vector \(v\). Find (a) the two-step transition matrix and (b) the distribution vectors after one, two, and three steps. $$ P=\left[\begin{array}{cc} 1 / 2 & 1 / 2 \\ 1 & 0 \end{array}\right], v=\left[\begin{array}{ll} 2 / 3 & 1 / 3 \end{array}\right] $$

If two indistinguishable dice are rolled, what is the probability of the event \(\\{(4,4),(2,3)\\} ?\) What is the corresponding event for a pair of distinguishable dice? HINT [See Example 2.]

Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, three green ones, two white ones, and one purple one. She grabs five of them. Find the probabilities of the following events, expressing each as a fraction in lowest terms. HINT [See Example 1.] She does not have all the green ones.

An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the coins and dice are distinguishable and fair, and that what is observed are the faces or numbers uppermost. Two dice are rolled; the numbers add to 9 .

Suppose two dice (one red, one green) are rolled. Consider the following events. A: the red die shows \(1 ; B:\) the numbers add to \(4 ; C:\) at least one of the numbers is \(1 ;\) and \(D:\) the numbers do not add to 11. Express the given event in symbols and say how many elements it contains. HINT [See Example 5.] The red die shows 1 and the numbers add to \(4 .\)
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