4.1: Introduction to Probability

The world is full of uncertainties. What are the chances that a student will pass a class? What are the chances that the price of a stock will go up? In this section we will introduce a framework which lays the foundation for the future study of statistics.

To quantify the uncertainties around us we use probability as a measure of how likely something is to happen. The more likely it is to happen the higher its probability, and the less likely something to happen the less its probability. We have to start with a few definitions.

Definition: Experiment

An experiment is an action whose result is not certain.

Here are some examples of actions whose outcomes are not certain:

Experiments
Flipping a coin
Flipping a coin twice (or flipping two distinct coins simultaneously)
Flipping a coin thrice (or flipping three distinct coins simultaneously)
Rolling a die
Rolling a die twice (or rolling a pair of dice simultaneously)
Rolling a die thrice (or rolling three distinct dice simultaneously)
Drawing a card from a standard 52-card deck
Spinning a roulette wheel

Definition: Simple outcome

Every experiment results in some simple outcome.

Here are some examples of simple outcomes for each of the experiments:

Experiments	Simple Outcome
Flipping a coin	H
Flipping a coin twice	HT
Flipping a coin thrice	TTH
Rolling a die	3
Rolling a die twice	2-4
Rolling a die thrice	5-1-4
Drawing a card	QH
Spinning a roulette	29

Definition: Sample space

The set of all outcomes of an experiment is called the sample space of the experiment.

Here are the sample spaces for each of the experiments:

Experiments	Simple Outcome	Sample Space
Flipping a coin	H	{H, T}
Flipping a coin twice	HT	{HH, HT, TH, TT}
Flipping a coin thrice	TTH	{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Rolling a die	3	{1,2,3,4,5,6}
Rolling a die twice	2-4	{1-1,1-2,1-3, etc.}
Rolling a die thrice	5-1-4	{1-1-1,1-1-2,1-1-3, etc.}
Drawing a card	QH	{2H,3H,4H,5H, etc.}
Spinning a roulette	29	{00,0,1,2,3,4, etc.}

Not every sample space is small enough that we can list all outcomes, but when it is the case, we need to know how to efficiently write the entire sample space.

For tossing coins there is a distinct pattern.

• When tossing a coin, we can get either Heads or Tails
• When tossing two coins, we can get for the first toss either Heads or Tails but now it can be combined with either Heads or Tails for the second toss.
• When tossing three coins, we can get for the first two tosses the same pattern as we just listed for tossing two coins but now this pattern can be combined with either Heads or Tails for the third toss.

For rolling dice there is another pattern.

• When rolling a die, the outcome is the number of dots that can be any number 1 through 6.
• When rolling two dice, the outcome is a pair of numbers that represent the number of dots on die 1 and die 2. There are 36 possible outcomes and the best way to list them is to organize it in the table.
• There is not enough space here to list ALL 216 possible outcomes for rolling three dice, but we can easily imagine what each outcome looks like.

When drawing a card from a standard 52-card deck, there are 52 possible outcomes – one for each distinct card in the deck:

When spinning an American roulette, there are 38 possible outcomes – one for each of the numbers 1 through 36 and 0 and 00. The best way to visualize it is to see the roulette table:

Definition: Event

An event is a collection of outcomes from the sample space.

For example, consider rolling a die for which the sample space is {1,2,3,4,5,6}. Any collection of simple outcomes from the list will define an event. Here are a few examples of sets created by including some of the outcomes from the sample space:

Event	Set Description
\(A\)	{2, 4, 6}
\(B\)	{1, 3, 5}
\(C\)	{5, 6}
\(D\)	{1, 2, 3, 4, 5, 6}
\(E\)	{ }

In probability we use the capital letters from the beginning of the alphabet to denote the events - A, B, C, D, E.

Some events may have a nice verbal description in English. Which of the listed events can be verbally described as “rolling an even number”? Naturally, we would say that the list {2,4,6} matches that description. Similarly, we can label the other events in the following way:

Event	Verbal Description	Set Description
\(A\)	Roll an even number	{2, 4, 6}
\(B\)	Roll an odd number	{1, 3, 5}
\(C\)	Roll a number greater than or equal to 5	{5, 6}
\(D\)	Roll a positive number	{1, 2, 3, 4, 5, 6}
\(E\)	Roll a number greater than 7	{ }

However, the majority of events will not have such a description. There are exactly sixty-four events that can be defined from the sample space of size 6. Most of them will not have an intuitive verbal description, e.g., try to come up with a verbal description of an event \(F=\{2, 3, 5\}\)? This is why it is important to learn how to work with the definition of an event rather than rely on its verbal representation.

Definition: Event occurred vs didn't occur

After finishing the experiment, we say an event occurred if the outcome of an experiment is in the collection that defines an event. We say an event did not occur if the outcome of an experiment is not in the collection that defines an event.

For each event and each outcome, we can decide whether an event has occurred or not. For example, when you roll 1 we say B and D occurred and A, C, E didn’t occur. Similarly, for event A we can say that it occurs when 2,4, or 6 are rolled and doesn’t occur when 1,3, or 5 are rolled. The rest of the table looks like this:

Event	Result is 1	Result is 2	Result is 3	Result is 4	Result is 5	Result is 6
\(A=\{2, 4, 6\}\)	X	V	X	V	X	V
\(B=\{1, 3, 5\}\)	V	X	V	X	V	X
\(C=\{5, 6\}\)	X	X	X	X	V	V
\(D=\{1, 2, 3, 4, 5, 6\}\)	V	V	V	V	V	V
\(E=\{ \}\)	X	X	X	X	X	X

It is important to understand why and when the event occurs and doesn’t occur depending on the outcome of the experiment.

Try It Yourself! \(\PageIndex{1}\)

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Try It Yourself! \(\PageIndex{5}\)

Try It Yourself! \(\PageIndex{6}\)

Try It Yourself! \(\PageIndex{7}\)

Try It Yourself! \(\PageIndex{8}\)

Notice that in the ongoing example event \(D=\{1, 2, 3, 4, 5, 6\}\) is the entire sample space. As strange as it may look event \(D\) satisfies the definition and therefore has to be treated as an event. Since \(D\) occurs for any outcome of the experiment, we call \(D\) a certain event. Now, take a look at event \(E=\{ \}\) which is the empty set. As strange as it may look event \(E\) also satisfies the definition and therefore has to be treated as an event. Since \(E\) occurs for no outcome of the experiment, we call \(E\) an impossible event. Let me remind you again that this is a small sample of events from all sixty-four possible events.

Hopefully by now you can see the importance of knowing how to work with the set description. Alternatively, one can think of an event as something that someone can make a bet on happening. For example, when you throw a die, someone can walk in and say I bet the number will be less than 4, and we know exactly which outcomes will win the bet and which outcomes will lose the bet.

Now, once we have the precise definition and informal definition of an event, we can define the probability of an event.

Definition: Probability

Recall that probability is a measure of how likely an event is to occur. We denote the probability of an event \(E\) as \(P(E)\). We define the probability of an impossible event to be equal to zero, and the probability of a certain event to be equal to one. Nothing can be more likely than a certain event and nothing can be less likely than an impossible event, therefore the following inequality must be true for any event \(E\):

\(0\leq P(E) \leq 1\)

Next, we will discuss different ways to compute the probability of an event.