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Chapter 7: Problem 27

An experiment consists of tossing a coin, rolling a die, and observing the outcomes. a. Describe an appropriate sample space for this experiment. b. Describe the event "a head is tossed and an even numher is rolled."

Short Answer

Expert verified
a) The appropriate sample space for this experiment is: \( S = \lbrace (H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6) \rbrace \) b) The event "a head is tossed and an even number is rolled" can be represented as the set: \( E = \lbrace (H,2), (H,4), (H,6) \rbrace \)

Step by step solution

01

a) Describe an appropriate sample space for this experiment

To describe an appropriate sample space for this experiment, we need to list all possible outcomes of both the coin toss and the die roll. For a coin toss, there are two possible outcomes: Heads (H) and Tails (T). For a die roll, there are six possible outcomes: 1, 2, 3, 4, 5, and 6. Since the two actions are independent, we can combine the two sets of outcomes to create a sample space of all possible outcomes for the experiment. To find the combined outcomes, we pair each outcome of the coin toss with each outcome of the die roll. The sample space will have a total of 2 (coin outcomes) x 6 (die outcomes) = 12 possible outcomes. Thus, the appropriate sample space for this experiment is: S = {(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}
02

b) Describe the event "a head is tossed and an even number is rolled"

To describe the event "a head is tossed and an even number is rolled", we need to identify the outcomes in the sample space that satisfy this condition. An even number is a number that is divisible by 2, and for the die roll, the possible even numbers are 2, 4, and 6. We are also given the condition that the coin toss results in a head. With this information, we can locate the outcomes in our sample space that match these criteria. These outcomes are: 1 - (H,2): A head is tossed, and a 2 is rolled. 2 - (H,4): A head is tossed, and a 4 is rolled. 3 - (H,6): A head is tossed, and a 6 is rolled. So, the event "a head is tossed and an even number is rolled" can be represented as the set: E = {(H,2), (H,4), (H,6)}

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Key Concepts

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

Outcomes in Probability
In probability, outcomes are the foundation of any experiment. An outcome is a possible result that can occur during a probability experiment. For instance, when you toss a coin, there are two outcomes you may experience: Heads (H) or Tails (T). Additionally, when you roll a die, the outcomes expand to six numbers: 1, 2, 3, 4, 5, and 6.
These outcomes are fundamental building blocks in determining the overall sample space of an experiment, which is a complete list of all possible outcomes. Without clearly identifying all outcomes, you cannot perform proper probability calculations or create meaningful event descriptions.
Understanding a Probability Experiment
A probability experiment is a procedure that can be repeated and leads to one of several distinct outcomes. Each time you perform this experiment, you might encounter a different result based on chance. For example, the experiment could be as simple as flipping a coin or as complex as conducting a series of clinical trials.
When defining a probability experiment like tossing a coin and rolling a die, both actions are considered independent. This means the outcome of the coin toss does not affect the outcome of the die roll. Therefore, we combine the results from each independent action to form a broader sample space. In our scenario, this experiment gives us a total of 12 possible pairings of outcomes.
  • Coin outcomes: Heads (H), Tails (T)
  • Die outcomes: 1, 2, 3, 4, 5, 6
These independent results together form the complete sample space from which we can explore further events.
Event Description in Probability
In probability, an event is a specific set of outcomes that satisfy a particular condition within the sample space. Describing an event involves identifying which outcomes meet the criteria set by the event.
Consider the event described as "a head is tossed and an even number is rolled." This event specifically requests both a head result from the coin and an even number from the die. We then look at the sample space and isolate the outcomes that fit this description:
  • (H,2): A head, with an even roll of 2
  • (H,4): A head, with an even roll of 4
  • (H,6): A head, with an even roll of 6
These three outcomes constitute the complete event description and highlight how a specific scenario can be defined within a probability framework.
Combinatorial Analysis in Probability
Combinatorial analysis involves determining how to count combinations of outcomes in a probability experiment. This area of study aids in understanding and calculating probabilities by organizing and simplifying complex sets of possible results.
For instance, when tasked with identifying the sample space for tossing a coin and rolling a die, we use combinatorial analysis to calculate the number of possible outcomes. Since the coin has 2 outcomes and the die has 6, we multiply these to obtain 12 total outcomes. This concept helps manage larger and more intricate sets of possibilities, ensuring that every result is accounted for.
By applying combinatorial methods, it becomes easier to manage and analyze the complexity of outcomes in probability. This allows for accurate prediction and understanding of the likelihood of various events occurring.

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Most popular questions from this chapter

The estimated probability that a brand-A, a brand-B, and a brand-C plasma TV will last at least \(30,000 \mathrm{hr}\) is \(.90, .85\), and \(.80\), respectively. Of the 4500 plasma TVs that Ace TV sold in a certain year, 1000 were brand A, 1500 were brand \(\mathrm{B}\), and 2000 were brand \(\mathrm{C}\). If a plasma TV set sold by Ace TV that year is selected at random and is still working after \(30,000 \mathrm{hr}\) of use a. What is the probability that it was a brand-A TV? b. What is the probability that it was not a brand-A TV?

If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, what is the probability of being dealt the given hand? A flush (but not a straight flush)

Let \(S=\left\\{s_{1}, s_{2}, s_{3}, s_{4}, s_{5}\right\\}\) be the sample space associated with an experiment having the following probability distribution: $$\begin{array}{llllll} \hline \text { Outcome } & s_{1} & s_{2} & s_{3} & s_{4} & s_{5} \\ \hline \text { Probability } & \frac{1}{14} & \frac{3}{14} & \frac{6}{14} & \frac{2}{14} & \frac{2}{14} \\ \hline \end{array}$$ Find the probability of the event: a. \(A=\left\\{s_{1}, s_{2}, s_{4}\right\\}\) b. \(B=\left\\{s_{1}, s_{5}\right\\}\) c. \(C=S\)

According to a study of 100 drivers in metropolitan Washington D.C. whose cars were equipped with cameras with sensors, the distractions and the number of incidents (crashes, near crashes, and situations that require an evasive maneuver after the driver was distracted) caused by these distractions are as follows: $$ \begin{array}{lccccccccc} \hline \text { Distraction } & A & B & C & D & E & F & G & H & I \\ \hline \text { Driving Incidents } & 668 & 378 & 194 & 163 & 133 & 134 & 111 & 111 & 89 \\ \hline \end{array} $$ where \(A=\) Wireless device (cell phone, PDA) $$ \begin{aligned} B &=\text { Passenger } \\ C &=\text { Something inside } \mathrm{car} \\ D &=\text { Vehicle } \\ E &=\text { Personal hygiene } \\ F &=\text { Eating } \\ G &=\text { Something outside car } \\ H &=\text { Talking/singing } \\ I &=\text { Other } \end{aligned} $$ If an incident caused by a distraction is picked at random, what is the probability that it was caused by a. The use of a wireless device? b. Something other than personal hygiene or eating?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) and \(B\) are mutually exclusive and \(P(B) \neq 0\), then \(P(A \mid B)=0 .\)
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