3.1 The Terminology of Probability

Everyday, decisions are made that involve uncertainty about the outcome. The ability to estimate and understand probability helps us make good decisions. Probability is a numerical measure that is associated with how certain we are of the outcomes of a particular experiment or activity. Examples of probability used in everyday life include the probability that it will rain today and the probability of winning the lottery.

An experiment is a planned operation carried out under controlled conditions. If the result is not predetermined, then the experiment is said to be a chance experiment. An experiment is any activity where the outcome is uncertain. Flipping a coin, rolling a pair of dice, or drawing a card from a deck of cards are all examples of an experiment.

A result of an experiment is called an outcome. For example, in the experiment of flipping a coin, a possible outcome is getting heads. The sample space of an experiment is the set of all possible outcomes of that experiment.  Three ways to represent a sample space are to list the possible outcomes, to create a tree diagram, or to create a Venn diagram. The uppercase letter [latex]S[/latex] is used to denote the sample space. For example, in the experiment of flipping a coin, the sample space has two outcomes:  heads or tails. In the notation of probability, we would write the sample space of flipping a coin like [latex]S= \{H, T\}[/latex] where [latex]H[/latex] is heads and [latex]T[/latex] is tails.

An event is any combination of outcomes. Generally, an event is a collection of outcomes that possess some trait or characteristic. Upper case letters like [latex]A[/latex] and [latex]B[/latex] are used to represent events. For example, if the experiment is to flip a coin two times, event [latex]A[/latex] might be getting at most one head in the two flips. In probability, we are interested in finding the probability of an event. The probability of an event [latex]A[/latex] is written [latex]P(A)[/latex].

Probability is a numerical measure of the likelihood that an event will occur. The probability of an event is the long-term relative frequency of that event. Probabilities are numbers between zero and one, inclusive—that is, zero, one, and all numbers between these values. Probabilities can be written as fractions, decimals, or percents.  [latex]P(A) = 0[/latex] means the event [latex]A[/latex] can never happen—the probability is [latex]0\%[/latex]. [latex]P(A) =1[/latex] means the event [latex]A[/latex] always happens—the probability is [latex]100\%[/latex]. [latex]P(A) = 0.5[/latex] means the event [latex]A[/latex] is equally likely to occur or not to occur—there is a [latex]50\%[/latex] chance [latex]A[/latex] will happen, and a [latex]50\%[/latex] chance [latex]A[/latex] will not happen.

Approaches to Determining Probability

The way that we calculate the probability of an event depends on the situation we are analyzing.

Classical Method Approach to Probability

Most often associated with games of chance, the classical method approach requires us to know that the outcomes of an experiment are equally likely to occur. We have already seen an experiment where the outcomes are equally likely to occur—flipping a coin. Equally likely means that each outcome of an experiment occurs with equal probability. In the experiment of tossing a fair coin, we know that we have a 50% chance of getting heads and a 50% chance of getting tails—the outcomes of heads or tails are equally likely to occur. If we roll a fair, six-sided die, we know that we have the same chance [latex]\left(\frac{1}{6}\right)[/latex] of getting any of the six faces—the outcomes of 1, 2, 3, 4, 5, 6 are equally likely to occur.

To calculate the probability of an event [latex]A[/latex] when all outcomes in the sample space are equally likely, count the number of outcomes for event [latex]A[/latex] and divide by the total number of outcomes in the sample space.

[latex]\begin{eqnarray*}\\P(A)&=&\frac{\text{number of outcomes in event }A}{\text{total number of outcomes in the sample space}}\\\\\end{eqnarray*}[/latex]

It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be unfair or biased. Two math professors in Europe had their statistics students test the Belgian one Euro coin and discovered that in [latex]250[/latex] trials, a head was obtained [latex]56\%[/latex] of the time and a tail was obtained [latex]44\%[/latex] of the time. The data seem to show that the coin is not a fair coin, but more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a board game. The spots on each face are usually small holes carved out and then painted to make the spots visible. The dice may or may not be biased because it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos make a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces and the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur.

Empirical Method Approach to Probability

The empirical or relative frequency approach to probability uses results from identical previous experiments that have been performed many times. Probabilities are based on historical or previously recorded data by determining the proportion of times an event occurs within the data. For example, a retail business owner might want to know the probability that a customer spends more than [latex]\$50[/latex] at their store. To determine this probability, the business owner would look at previous sales, count the number of sales over [latex]\$50[/latex] and then divide that number by the total number of previous sales.

To calculate an empirical probability, repeat the experiment over a large number of trials and record the result of each trial. To find the probability of event [latex]A[/latex], count the number of times event [latex]A[/latex] happened and divide by the total number of trials.

[latex]\begin{eqnarray*}\\P(A)=\frac{\text{number of times }A\text{ occurs}}{\text{total number of trials}}\\\\\end{eqnarray*}[/latex]

To get an accurate probability using this approach, it is important that the experiment is repeated a very large number of times. This important characteristic of probability experiments is known as the law of large numbers, which states that as the number of repetitions of an experiment increases, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes do not happen according to any set pattern or order, overall, the long-term observed relative frequency will approach the theoretical probability.

Subjective Method Approach to Probability

In the subjective method approach to probability, probabilities are determined by educated guess, personal belief, intuition, or expert reasoning. A subjective probability is essentially a guess, but a guess based on an accumulation of knowledge, understanding, and experience. Estimating the probability the price of a stock goes down over time or the probability a certain sports team will win a championship are examples of subjective probability.

Video: “Probability: Tossing 2 Coins (Head/Tail)” by Joshua Emmanuel [5:56] is licensed under the Standard YouTube License.Transcript and closed captions available on YouTube.

Exercises

1. A box is filled with several party favours. It contains [latex]12[/latex] hats, [latex]15[/latex] noisemakers, [latex]10[/latex] finger traps, and [latex]5[/latex] bags of confetti. A party guest randomly selects one of the party favours from the box.
   1. Find the probability of getting a hat.
   2. Find the probability of getting a noisemaker.
   3. Find the probability of getting a finger trap.
   4. Find the probability of getting a bag of confetti.
   Click to see Answer

   1. [latex]0.2857[/latex]
   2. [latex]0.3571[/latex]
   3. [latex]0.2381[/latex]
   4. [latex]0.119[/latex]

    

2. A jar of [latex]150[/latex] jelly beans contains [latex]22[/latex] red jelly beans, [latex]38[/latex] yellow, [latex]20[/latex] green, [latex]28[/latex] purple, [latex]26[/latex] blue, and the rest are orange. A jelly bean is selected at random.
   1. Find the probability of getting a blue jelly bean.
   2. Find the probability of getting a green jelly bean.
   3. Find the probability of getting a purple jelly bean.
   4. Find the probability of getting a red jelly bean.
   5. Find the probability of getting a yellow jelly bean.
   6. Find the probability of getting an orange jelly bean.
   Click to see Answer

   1. [latex]0.1733[/latex]
   2. [latex]0.1333[/latex]
   3. [latex]0.1867[/latex]
   4. [latex]0.1467[/latex]
   5. [latex]0.2533[/latex]
   6. [latex]0.1067[/latex]

    

3. Suppose a card is drawn from a standard deck of [latex]52[/latex] cards.
   1. What is the probability the card is red?
   2. What is the probability the card is a club?
   3. What is the probability the card is a face card (jack, queen, or king)?
   4. What is the probability the card is an ace?
   5. What is the probability the card is the jack of hearts?
   Click to see Answer

   1. [latex]0.5[/latex]
   2. [latex]0.25[/latex]
   3. [latex]0.2308[/latex]
   4. [latex]0.0769[/latex]
   5. [latex]0.0192[/latex]

    

4. Suppose a fair, six-sided die is rolled.
   1. What is the probability of rolling an even number?
   2. What is the probability of rolling a [latex]2[/latex], [latex]3[/latex], or[latex]5[/latex]?
   3. What is the probability of rolling a [latex]3[/latex] or a [latex]6[/latex]?
   Click to see Answer

   1. [latex]0.5[/latex]
   2. [latex]0.5[/latex]
   3. [latex]0.3333[/latex]

    

5. What is the word for the set of all possible outcomes?
   
   Click to see Answer

   sample space.

    

6. A sample of students was surveyed and asked how many movies they watched last week. The results are given in the table below.
   

   Number of Movies	Frequency
   0	5
   1	9
   2	6
   3	4
   4	1

   1. What is the probability a student watched [latex]0[/latex] movies last week?
   2. What is the probability a student watched at most [latex]1[/latex] movie last week?
   3. What is the probability a student watched [latex]2[/latex] or more movies last week?
   Click to see Answer

   1. [latex]0.2[/latex]
   2. [latex]0.56[/latex]
   3. [latex]0.44[/latex]

    

7. A sample of students was surveyed and asked how many pairs of sneakers they own. The results are given in the table below.
   

   Number of Pairs of Sneakers	Frequency
   1	2
   2	5
   3	8
   4	12
   5	12
   6	0
   7	1

   1. What is the probability a student owns exactly [latex]2[/latex] pairs of sneakers?
   2. What is the probability a student owns exactly [latex]6[/latex] pairs of sneakers?
   3. What is the probability a student owns  [latex]4[/latex] or more pairs of sneakers?
   4. What is the probability a student owns [latex]3[/latex] or fewer pairs of sneakers?
   5. What is the probability a student owns [latex]5[/latex] or [latex]7[/latex] pairs of sneakers?
   Click to see Answer

   1. [latex]0.125[/latex]
   2. [latex]0[/latex]
   3. [latex]0.625[/latex]
   4. [latex]0.375[/latex]
   5. [latex]0.325[/latex]

    

“3.2 The Terminology of Probability” and “3.8 Exercises” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.