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Chapter 4: Problem 24

Four coins are tossed. How many simple events are in the sample space?

Short Answer

Expert verified
Answer: There are 16 simple events in the sample space when four coins are tossed.

Step by step solution

01

Understand the problem

Four coins are tossed, and each coin has two possible outcomes - heads (H) or tails (T). We need to determine the total number of ways these outcomes can be combined for all four coins.
02

Calculate the number of simple events

To find the total number of simple events, we can use the counting principle. The counting principle states that if there are n outcomes for event A and m outcomes for event B, then there are n * m possible outcomes for both events combined. In our case, each coin has two possible outcomes (H or T), so the total number of simple events can be calculated as the product of the number of outcomes for each coin: 2 * 2 * 2 * 2
03

Compute the final result

Multiply the number of outcomes for each coin to find the total number of simple events in the sample space: 2 * 2 * 2 * 2 = 16 There are 16 simple events in the sample space when four coins are tossed.

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Key Concepts

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

Counting Principle
The Counting Principle is an essential concept in probability and combinatorics. It provides a way to determine the total number of possible outcomes by multiplying the number of choices available at each stage of a process.
For example, when tossing a set of coins, each coin can land on either heads or tails. The principle assists in calculating the total combinations of these events without listing each possibility.
  • Imagine you toss a single coin. It has 2 outcomes: heads (H) and tails (T).
  • If you toss two coins, each coin maintains 2 outcomes, so you multiply these: 2 (first coin) 脳 2 (second coin) = 4 outcomes.
  • This logic extends to four coins, multiplying 2 for each coin tossed: 2 脳 2 脳 2 脳 2 = 16 outcomes.
The Counting Principle scales efficiently no matter how many coins, or stages, are involved, providing a swift method to manage complex calculations of outcomes.
Simple Events
In probability, a simple event is any single occurrence within a sample space with a specific outcome.
Think of it this way: when tossing one coin, a simple event could be getting heads or tails. These are distinct possibilities that occur as individual outcomes.
  • With four tossed coins, each specific sequence of results (like HHTT or THHT) is a simple event.
  • Individual simple events don鈥檛 overlap or share outcomes with other events.
  • The collection of all simple events for the tossed coins describes the entire sample space.
Simple events are building blocks in probability, helping to construct and understand the broader picture of possible outcomes.
Outcomes in Probability
Outcomes in probability refer to the possible results that can occur from a random experiment, such as tossing coins, rolling dice, or drawing cards.
Identifying outcomes is crucial for creating a complete picture of what might happen in a given probabilistic scenario.
  • The totality of these possible outcomes is known as the sample space.
  • For four coins, each having two results (H or T), outcomes form various combinations leading to 16 outcomes.
  • The whole set of outcomes, like HHHH, HHHT, etc., defines the sample space for the experiment.
In understanding probability, recognizing and organizing outcomes helps quantify and assess how likely different events are to occur.

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Most popular questions from this chapter

If you toss a pair of dice, the sum \(T\) of the numbers appearing on the upper faces of the dice can assume the value of an integer in the interval \(2 \leq T \leq 12\) a. Find the probability distribution for \(T\). Display this probability distribution in a table. b. Construct a probability histogram for \(P(T)\). How would you describe the shape of this distribution?

You have three groups of distinctly different items, four in the first group, seven in the second, and three in the third. If you select one item from each group, how many different triplets can you form?

Suppose \(5 \%\) of all people filing the long income tax form seek deductions that they know are illegal, and an additional \(2 \%\) incorrectly list deductions because they are unfamiliar with income tax regulations. Of the \(5 \%\) who are guilty of cheating, \(80 \%\) will deny knowledge of the error if confronted by an investigator. If the filer of the long form is confronted with an unwarranted deduction and he or she denies the knowledge of the error, what is the probability that he or she is guilty?

A survey to determine the availability of flextime schedules in the California workplace provided the following information for 220 firms located in two California cities. $$\begin{array}{cccc}\multicolumn{4}{c} {\text { Flextime Schedule }} \\\\\hline \text { City } & \text { Available } & \text { Not Available } & \text { Total } \\\\\hline A & 39 & 75 & 114 \\\B & 25 & 81 & 106 \\\\\hline \text { Totals } & 64 & 156 & 220\end{array}$$ A company is selected at random from this pool of 220 companies. a. What is the probability that the company is located in city \(A\) ? b. What is the probability that the company is located in city \(B\) and offers flextime work schedules? c. What is the probability that the company does not have flextime schedules? d. What is the probability that the company is located in city \(B\), given that the company has flextime schedules available?

A slot machine has three slots; each will show a cherry, a lemon, a star, or a bar when spun. The player wins if all three slots show the same three items. If each of the four items is equally likely to appear on a given spin, what is your probability of winning?
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