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Chapter 6: Problem 3

Consider the chance experiment in which the type of transmission-automatic (A) or manual (M) - is recorded for each of the next two cars purchased from a certain dealer. a. What is the set of all possible outcomes (the sample space)? b. Display the possible outcomes in a tree diagram. c. List the outcomes in each of the following events. Which of these events are simple events? i. \(B\) the event that at least one car has an automatic transmission ii. \(C\) the event that exactly one car has an automatic transmission iii. \(D\) the event that neither car has an automatic transmission d. What outcomes are in the event \(B\) and \(C ?\) In the event \(B\) or \(C\) ?

Short Answer

Expert verified
The sample space of car transmission outcomes is {AA, AM, MA, MM}. The event that at least one car has an automatic includes {AA, AM, MA}, exactly one car with an automatic includes {AM, MA}, neither car having automatic has {MM}. Outcomes in both event B and C are {AA, AM, MA}, and for B or C are also {AA, AM, MA}.

Step by step solution

01

Identify Sample Space

Identify the possible outcomes when purchasing two cars each with either automatic \(A\) or manual \(M\) transmission. This leads to four outcomes overall: \(AA, AM, MA, MM\).
02

Construct a Tree Diagram

From the origin point, create two branches representing the options for the first vehicle: automatic \(A\) and manual \(M\). Branch off each of these options again for the second vehicle, again with automatic \(A\) and manual \(M\). This creates outcomes \(AA, AM, MA, MM\).
03

List Outcomes for Each Event

i. Event \(B\) (at least one automatic transmission) includes outcomes \(AA, AM, MA\).\nii. Event \(C\) (exactly one automatic transmission) includes outcomes \(AM, MA\).\niii. Event \(D\) (neither car has automatic transmission) includes the outcome \(MM\). Here, \(B\) and \(C\) are compound events, while \(D\) is a simple one as it only includes one outcome.
04

Define Overlaps in Events

Event \(B\) and \(C\) (both cars are automatic or exactly one is automatic) includes outcomes \(AA, AM, MA\). The event \(B\) or \(C\) (at least one car is automatic or one car is automatic) includes outcomes \(AA, AM, MA\).

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Key Concepts

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

Sample Space
In probability, a sample space is the set of all possible outcomes from a chance experiment. When we talk about the probability of certain events occurring, we're checking to see if specific conditions are met within this space. Let's dive into the specific exercise involving car transmissions.
The exercise involves identifying the type of transmission, either automatic (A) or manual (M), for two cars. Here, the sample space consists of all possible combinations of these transmissions for both cars.
For an experiment like this, it's helpful to think about each car individually and then combine those outcomes. The possible outcomes for the first car - automatic or manual - can pair with similar outcomes for the second car. Thus, the sample space includes four combinations:
  • AA - both cars have automatic transmission.
  • AM - first car is automatic, second is manual.
  • MA - first car is manual, second is automatic.
  • MM - both cars have manual transmission.
Recognizing the sample space helps us calculate the probabilities of various events.
Tree Diagram
A tree diagram is an excellent tool for visually representing potential outcomes of a probability experiment. It helps you think through the sequencing of events and simplifies calculation of probabilities.
For our exercise, we use a tree diagram to map out the possible sequences of transmission types for two cars. Begin by drawing two branches, one for each possible outcome of the first car:
  • Automatic (A)
  • Manual (M)
From each of these branches, draw another two branches to represent the outcomes for the second car: automatic or manual. This results in the combined outcomes:
  • Start with A:
    • AA (both cars automatic)
    • AM (first automatic, second manual)
  • Start with M:
    • MA (first manual, second automatic)
    • MM (both cars manual)
Tree diagrams help you visually organize outcomes, making it easier to identify different events.
Simple and Compound Events
Understanding the difference between simple and compound events is crucial in probability. A simple event involves only one outcome from the sample space, whereas a compound event contains multiple outcomes.
In our example, we have events based on the combination of transmission types:
  • Event B: At least one car has an automatic transmission. This includes outcomes: AA, AM, MA. Since multiple outcomes satisfy this condition, it's a compound event.
  • Event C: Exactly one car has an automatic transmission. This involves outcomes: AM, MA, making it another compound event.
  • Event D: Neither car has an automatic transmission. This only involves one outcome: MM. Therefore, it is a simple event.
By distinguishing between these events, we get better insight into how probabilities operate in different situations. It helps in making predictions and understanding outcomes of real-world occurrences clearly.

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Most popular questions from this chapter

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