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Chapter 4: Problem 3

Use the mn Rule to find the number. Four coins are tossed. How many simple events are in the sample space?

Short Answer

Expert verified
Answer: 16

Step by step solution

01

Identify m and n in the mn Rule

In this problem, m represents the number of outcomes for each coin toss, which is 2 (heads or tails), and n represents the number of tosses, which is 4.
02

Apply the mn Rule

Now that we have identified m and n, we can apply the mn Rule to find the total number of simple events in the sample space. The mn Rule is: \(Number\:of\:simple\:events\:in\:sample\:space = m^n\) In this case, we have: \(Number\:of\:simple\:events\:in\:sample\:space = 2^4\)
03

Calculate the number of simple events

Calculate the value of \(2^4\): \(Number\:of\:simple\:events\:in\:sample\:space = 16\) There are 16 simple events in the sample space for this exercise 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.

Mn Rule
When talking about probability, the Mn Rule is a straightforward way to calculate the total number of possible outcomes, or simple events, of a series of tasks.
This comes in handy especially when events are independent, like tossing coins or rolling dice.
  • m: This represents the number of outcomes per event. For example, when tossing a coin, you get 2 outcomes: heads or tails.
  • n: This signifies the number of times a particular event takes place. For instance, if you toss the coin 4 times, then n is 4.
The Mn Rule formula is given by: \(m^n\).
This means that for each event, you multiply the outcomes. So, if a coin toss (m = 2 outcomes) is repeated 4 times (n = 4), the formula becomes \(2^4 = 16\).
This means you'll have 16 different ways the coins could land. This formula helps predict how tasks divide into possible outcomes efficiently.
Sample Space
In probability, a sample space is the set of all possible outcomes of an event. For instance, while tossing a coin, the sample space includes all results that might appear.
Let's break that down:
  • Each outcome, like getting a head or a tail on a single coin toss, is part of this set.
  • When multiple events occur, like tossing four coins, the sample space incorporates every possible combination of events.
This means if you toss a coin four times, the sample space consists of all sequences like HHHH, HHHT, HHTH, ... TTTT, and so on.
By understanding the sample space, you can predict the likelihood of each specific result. You know exactly how many outcomes are possible, which is a big win for solving probability problems.
Simple Events
A simple event is the most basic possible outcome of an experiment, which cannot be broken down any further.
Each distinct outcome, like getting a head once in a coin toss, represents a simple event.

To give a clearer picture:

  • In a single coin toss, the simple events are "head" and "tail."
  • For four tosses, each sequence of heads and tails, like HHTT or TTHH, is a simple event.
Simple events are fundamental in constructing a sample space since they represent each unique outcome.
These events help in determining probabilities because once you know how many simple events are in the sample space, you can calculate the likelihood of any particular outcome happening.

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Most popular questions from this chapter

Suppose that \(P(A)=.4\) and \(P(B)=.2 .\) If events \(A\) and \(B\) are independent, find these probabilities: a. \(P(A \cap B)\) b. \(P(A \cup B)\)

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?

In how many ways can you select two people from a group of 20 if the order of selection is not important?

Suppose that, in a particular city, airport \(A\) handles \(50 \%\) of all airline traffic, and airports \(B\) and \(C\) handle \(30 \%\) and \(20 \%,\) respectively. The detection rates for weapons at the three airports are \(.9, .8,\) and .85, respectively. If a passenger at one of the airports is found to be carrying a weapon through the boarding gate, what is the probability that the passenger is using airport \(A\) ? Airport \(C ?\)

Five cards are selected from a 52-card deck for a poker hand. a. How many simple events are in the sample space? b. A royal flush is a hand that contains the \(\mathrm{A}, \mathrm{K}, \mathrm{Q}\), \(\mathrm{J},\) and \(10,\) all in the same suit. How many ways are there to get a royal flush? c. What is the probability of being dealt a royal flush?
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