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QM Course Guide

What is Probability?

What is Probability?

“Probability is a measure that is associated with how certain we are of outcomes of a particular experiment or activity.”

Why is Understanding Probability Important?

Why is understanding probability in statistics important?

We use probability in our everyday lives to make choices.  Should I cross the street on a red light? Or, should I spend time working on my homework, or studying? 

Cartoon: 2 kids sitting outside the principal's office. One says to the other "I wish we hadn't learned probability 'cause I don't think our odds are good." 

In statistics probability is a formalized system that helps us understand the certainty of outcomes of experiments or activities.

The p-value, (Please LINK TO DEFINITION) a measure of probability, is also used in testing hypotheses (link “hypotheses” to section on hypothesis testing).  We take information from sample statistics and generalize about larger populations.

Basic of Probability

Probability Outcomes: The Sample Space

The sample space is the collection of all possible outcomes of an experiment. The outcomes are commonly expressed using one of the four modes illustrated below.



The p-value is the probability that you would obtain the answer you have, assuming the occurrence is random.

Measuring Probability

How do we measure probability?

We measure probability using the p-value. Where:

0 = no chance of an event happening, and

1 = the event is certain to happen.

Representing Probability

How do we represent probabilities?

We can represent probabilities as a fraction, a decimal or a percentage. Certainty ranges from 0 (no likelihood of an event occurring) to 1, or 100% (complete certainty). 

For example, what is the probability of each number on a six-sided die turning up when rolled?

Number on die







Probability expressed as a fraction

1 / 6

1 / 6

1 / 6

1 / 6

1 / 6

1 / 6

Probability expressed as a decimal







Probability expressed as a percent







Calculating Probability

Methods for Calculating Probability

Picture of a hand flipping a coinEmpirical / Relative Frequency is calculated from experiment results.

It is expressed as:       

P (A)   =   number of times A occurred

               total number of observations

For example:   Flip a coin 50 times.

You get: 23 heads & 27 tails.

The empirical probability of heads turning up is:       

23 / 50.

Theoretical probability is calculated when you use your powers of deduction to determine probability.

When all outcomes are equally likely we can easily calculate the theoretical probability of an event.

It is expressed as:        P (A) = number of outcomes of event A

                                                 total number of outcomes

For example:   Flip a coin 50 times.

                        There are two equally likely results, heads or tails (50%, ½, 0.5)


The theoretical probability of getting heads is half of 50, or 25.

Image of experimental versus theoretical probability. The 5 appears 140 times when dice rolled 1000 times versus 5 appears 1 out of 6 when rolled 1 time.

The Law of Large Numbers

This law states that the more often your repeat an experiment the closer the empirical / relative frequency comes to the theoretical probability.

Can we test this theory? Flip a coin.  Can we embed the link with the flippable coin?? If not we can jut keep linked word)

Do it 10 times.

Now do it 25 times.

Now do it 100 times.             

Now do it 1000 times… NO? Too much?           

 How many times did heads turn up? What about tails?  Do your results support the theory?

It can take many tries for the fraction of heads to approach 50%, as illustrated in the image on the commemorative stamp, above.   

Independent Events exist when the probability of one outcome does not affect another.

Cartoon Image of an angry bird getting ready to roll some dice at a casino table. Example: We are rolling 2 dice. We want to know the probability of rolling a 6 on each.

The probability of rolling a 6 on each die is 1/6. To calculate the probability of rolling a six on both we multiply.

P (a and b)       = P(a) x P(b)    

                        = 1/6  x 1/6 

                        = 1/36 (or 0.028, or 2.8%)     

Image of  an open soft bag with marbles poured outDependent Events exist when probability events are related


Example: We pull out a blue then a red marble from a bag of 20 marbles where there are 5 green, 5 red, 5 yellow and 5 blue.


 P (a) = 5 / 20

 P (a then b)     = p (a) * p (b given a)

                                    = ¼ * 5/ 19 

                                    = 5 / 76 

                                    = 0.066   or    6.6%                    

The Gambler's Fallacy

Watch out!

Sometimes mistaken beliefs lead us to misinterpret the world around us.

The Gambler’s Fallacy is the mistaken belief that a streak of bad luck makes a person “due” for a streak of good luck, or that a streak of good luck will continue. (Bennett, Briggs and Triola, 2018, p. 209)