Easy Way to Formulate Number of Ways an Event Can Occur

Learning Outcomes

• Describe a sample space and simple and compound events in it using standard notation
• Calculate the probability of an event using standard notation
• Calculate the probability of two independent events using standard notation
• Recognize when two events are mutually exclusive
• Calculate a conditional probability using standard notation

Probability is the likelihood of a particular outcome or event happening. Statisticians and actuaries use probability to make predictions about events.  An actuary that works for a car insurance company would, for example, be interested in how likely a 17 year old male would be to get in a car accident.  They would use data from past events to make predictions about future events using the characteristics of probabilities, then use this information to calculate an insurance rate.

In this section, we will explore the definition of an event, and learn how to calculate the probability of it's occurance.  We will also practice using standard mathematical notation to calculate and describe different kinds of probabilities.

Basic Concepts

If you roll a die, pick a card from deck of playing cards, or randomly select a person and observe their hair color, we 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
• The sample space is the set of all possible simple events.

example

If we roll a standard 6-sided die, describe the sample space and some simple events.

Basic Probability

Given that all outcomes are equally likely, we can compute the probability of an event E using this formula:

[latex]P(E)=\frac{\text{Number of outcomes corresponding to the event E}}{\text{Total number of equally-likely outcomes}}[/latex]

examples

If we roll a 6-sided die, calculate

1. P(rolling a 1)
2. P(rolling a number bigger than 4)

This video describes this example and the previous one in detail.

Let's say you have a bag with 20 cherries, 14 sweet and 6 sour. If you pick a cherry at random, what is the probability that it will be sweet?

Show Solution

There are 20 possible cherries that could be picked, so the number of possible outcomes is 20. Of these 20 possible outcomes, 14 are favorable (sweet), so the probability that the cherry will be sweet 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 cherries is the same as the probability of picking any other. This wouldn't be true if (let us imagine) the sweet cherries are smaller than the sour ones. (The sour cherries 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 cases, we rely heavily on the assumption of equal probability for all outcomes.

Try It

At some random moment, you look at your clock and note the minutes reading.

a. What is probability the minutes reading is 15?

b. What is the probability the minutes reading is 15 or less?

Cards

A standard deck of 52 playing cards consists of four suits (hearts, spades, diamonds and clubs). Spades and clubs are black while hearts and diamonds are red. Each suit contains 13 cards, each of a different rank: an Ace (which in many games functions as both a low card and a high card), cards numbered 2 through 10, a Jack, a Queen and a King.

example

Compute the probability of randomly drawing one card from a deck and getting an Ace.

This video demonstrates both this example and the previous cherry example on the page.

Certain and Impossible events

• An impossible event has a probability of 0.
• A certain event has a probability of 1.
• The probability of any event must be [latex]0\le P(E)\le 1[/latex]

Try It

In the course of this section, if you compute a probability and get an answer that is negative or greater than 1, you have made a mistake and should check your work.

Types of Events

Complementary Events

Now let us examine the probability that an event does not happen. As in the previous section, consider the situation of rolling a six-sided die and first compute the probability of rolling a six: the answer is P(six) =1/6. Now consider the probability that we do not roll a six: there are 5 outcomes that are not a six, so the answer is P(not a six) = [latex]\frac{5}{6}[/latex]. Notice that

[latex]P(\text{six})+P(\text{not a six})=\frac{1}{6}+\frac{5}{6}=\frac{6}{6}=1[/latex]

This is not a coincidence.  Consider a generic situation with n possible outcomes and an event E that corresponds to m of these outcomes. Then the remaining n – m outcomes correspond to E not happening, thus

[latex]P(\text{not}E)=\frac{n-m}{n}=\frac{n}{n}-\frac{m}{n}=1-\frac{m}{n}=1-P(E)[/latex]

Complement of an Event

The complement of an event is the event "E doesn't happen"

• The notation [latex]\bar{E}[/latex] is used for the complement of event E.
• We can compute the probability of the complement using [latex]P\left({\bar{E}}\right)=1-P(E)[/latex]
• Notice also that [latex]P(E)=1-P\left({\bar{E}}\right)[/latex]

example

If you pull a random card from a deck of playing cards, what is the probability it is not a heart?

This situation is explained in the following video.

Try It

Probability of two independent events

example

Suppose we flipped a coin and rolled a die, and wanted to know the probability of getting a head on the coin and a 6 on the die.

The prior example contained two independent events. Getting a certain outcome from rolling a die had no influence on the outcome from flipping the coin.

Independent Events

Events A and B are independent events if the probability of Event B occurring is the same whether or not Event A occurs.

example

Are these events independent?

1. A fair coin is tossed two times. The two events are (1) first toss is a head and (2) second toss is a head.
2. The two events (1) "It will rain tomorrow in Houston" and (2) "It will rain tomorrow in Galveston" (a city near Houston).
3. You draw a card from a deck, then draw a second card without replacing the first.

When two events are independent, the probability of both occurring is the product of the probabilities of the individual events.

P(A and B) for independent events

If events A and B are independent, then the probability of both A and B occurring is

[latex]P\left(A\text{ and }B\right)=P\left(A\right)\cdot{P}\left(B\right)[/latex]

where P(A and B) is the probability of events A and B both occurring, P(A) is the probability of event A occurring, and P(B) is the probability of event B occurring

If you look back at the coin and die example from earlier, you can see how the number of outcomes of the first event multiplied by the number of outcomes in the second event multiplied to equal the total number of possible outcomes in the combined event.

example

In your drawer you have 10 pairs of socks, 6 of which are white, and 7 tee shirts, 3 of which are white. If you randomly reach in and pull out a pair of socks and a tee shirt, what is the probability both are white?

Examples of joint probabilities are discussed in this video.

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The previous examples looked at the probability of both events occurring. Now we will look at the probability of either event occurring.

example

Suppose we flipped a coin and rolled a die, and wanted to know the probability of getting a head on the coin or a 6 on the die.

P(A or B)

The probability of either A or B occurring (or both) is

[latex]P(A\text{ or }B)=P(A)+P(B)–P(A\text{ and }B)[/latex]

example

Suppose we draw one card from a standard deck. What is the probability that we get a Queen or a King?

See more about this example and the previous one in the following video.

In the last example, the events were mutually exclusive, so P(A or B) = P(A) + P(B).

Try It

example

Suppose we draw one card from a standard deck. What is the probability that we get a red card or a King?

Try It

In your drawer you have 10 pairs of socks, 6 of which are white, and 7 tee shirts, 3 of which are white. If you reach in and randomly grab a pair of socks and a tee shirt, what the probability at least one is white?

Example

The table below shows the number of survey subjects who have received and not received a speeding ticket in the last year, and the color of their car. Find the probability that a randomly chosen person:

1. Has a red car and got a speeding ticket
2. Has a red car or got a speeding ticket.

	Speeding ticket	No speeding ticket	Total
Red car	15	135	150
Not red car	45	470	515
Total	60	605	665

This table example is detailed in the following explanatory video.

Try It

Conditional Probability

In the previous section we computed the probabilities of events that were independent of each other. We saw that getting a certain outcome from rolling a die had no influence on the outcome from flipping a coin, even though we were computing a probability based on doing them at the same time.

In this section, we will consider events thataredependent on each other, called conditional probabilities.

Conditional Probability

The probability the event B occurs, given that event A has happened, is represented as

P(B | A)

This is read as "the probability of B given A"

For example, if you draw a card from a deck, then the sample space for the next card drawn has changed, because you are now working with a deck of 51 cards. In the following example we will show you how the computations for events like this are different from the computations we did in the last section.

example

What is the probability that two cards drawn at random from a deck of playing cards will both be aces?

Conditional Probability Formula

If Events A and B are not independent, then

P(A and B) = P(A) · P(B | A)

example

If you pull 2 cards out of a deck, what is the probability that both are spades?

Try It

Example

The table below shows the number of survey subjects who have received and not received a speeding ticket in the last year, and the color of their car. Find the probability that a randomly chosen person:

1. has a speeding ticket given they have a red car
2. has a red car given they have a speeding ticket

Speeding ticket	No speeding ticket	Total
Red car	15	135	150
Not red car	45	470	515
Total	60	605	665

These kinds of conditional probabilities are what insurance companies use to determine your insurance rates. They look at the conditional probability of you having accident, given your age, your car, your car color, your driving history, etc., and price your policy based on that likelihood.

View more about conditional probability in the following video.

Example

If you draw two cards from a deck, what is the probability that you will get the Ace of Diamonds and a black card?

These two playing card scenarios are discussed further in the following video.

Try It

Example

A home pregnancy test was given to women, then pregnancy was verified through blood tests.  The following table shows the home pregnancy test results.

Find

1. P(not pregnant | positive test result)
2. P(positive test result | not pregnant)

	Positive test	Negative test	Total
Pregnant	70	4	74
Not Pregnant	5	14	19
Total	75	18	93

See more about this example here.

Try It

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Source: https://courses.lumenlearning.com/wmopen-mathforliberalarts/chapter/computing-the-probability-of-an-event/

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