Question

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

Short answer

Answer: 16

Step-by-step solution

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.

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: \(Number\:of\:simple\:events\:in\:sample\:space = 2^4\)

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.

Key concepts

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.