Chapter 6 Probability
6.1 Concepts of Probability
Learning Objectives
By the end of this section, you will be able to:
- Identify a simple event, a compound event, and the sample space of an experiment
- Compute probabilities using the basic probability formula
- Identify complementary events and compute their probabilities
- Compute the odds of an event
- Identify independent events and compute their probabilities
- Compute the probability of an “or” statement
If you roll a die, pick a card from a deck of playing cards, or randomly select a person and observe their hair color, you are executing an experiment or procedure. In probability, we look at the likelihood of different outcomes. We begin with some terminology.
Events and Outcomes
The result of an experiment is called an outcome.
An event is any particular outcome or group of outcomes.
A simple event is an event that cannot be broken down further.
A simple event consists of exactly one outcome.
The sample space is the set of all possible simple events.
Example 1
If we roll a standard 6-sided die, describe the sample space and some simple events.
The sample space is the set of all possible simple events: {1, 2, 3, 4, 5, 6}
Some examples of simple events:
We roll a 1.
We roll a 5.
Some compound events:
We roll a number bigger than 4.
We roll an even number.
Basic Probability
Given that all outcomes are equally likely, we can compute the probability of an event [latex]E[/latex] using this formula:
[latex]P(E) =[/latex] [latex]\frac{\text { number of outcomes corresponding to event E}}{\text { total number of equally likely outcomes}}[/latex]
Example 2
If we roll a 6-sided die, calculate:
a) [latex]P(\text{rolling a 1})[/latex]
Recall that the sample space is {1, 2, 3, 4, 5, 6}. There is one outcome corresponding to “rolling a 1,” so the probability is [latex]\frac{1}{6}[/latex].
b) [latex]P(\text{rolling a number bigger than 4})[/latex]
There are two outcomes bigger than a 4, so the probability is [latex]\frac{2}{6} = \frac{1}{3}[/latex] .
Probabilities are essentially fractions and can be reduced to lower terms like fractions.
Example 3
Let’s say you have a bag with 20 figs, 14 ripe and 6 not-quite-ripe. If you pick a fig at random, what is the probability that it will be ripe?
There are 20 possible figs that could be picked, so the number of possible outcomes is 20. Of these 20 possible outcomes, 14 are favorable (ripe), so the probability that the fig will be ripe is [latex]\frac{14}{20} = \frac{7}{10}[/latex].
There is one potential complication to this example, however. It must be assumed that the probability of picking any of the figs is the same as the probability of picking any other. This wouldn’t be true if (let us imagine) the ripe figs are smaller than the not-quite-ripe ones. (The not-quite-ripe figs would come to hand more readily when you sampled from the bag.) Let us keep in mind, therefore, that when we assess probabilities in terms of the ratio of favorable to all potential