What isProbability?

The Mathematics of Chance

• How many possible outcomes are there with a single 6-sided die?

• What are your “chances” of rolling a 6?

• Can we generalize what you just did?

The Origins of Probability TheoryBlaise Pascal (1623-1662)

Pierre Fermat (1601-1665)

The Gambler’s Dispute…

"A gambler's dispute … a game consisted in throwing a pair of dice 24 times; the problem was to decide whether or not to bet even money on the occurrence of at least one "double six" during the 24 throws.

The gambler’s dispute (1654)

• This famous dispute led to the formal development of the mathematical theory of probability

Let’s simulate this…• How many possible

outcomes are there?• What fraction of these is

a “double-six”?• How can we quantify the

odds?• How many times would

expect to get 6-6 in 24 tries?

• How likely would it be to play this game 36 times and NOT get 6-6?

You have a 36% chance of not getting 6-6 in 36 throws (1:2 odds)

Link to Excel simulation

Happy Birthday!

• What is the probability that two of you share the same birthday?

• There are 40 people in class – would you rate the chances as 50/50, better or worse?

• Let’s test this!Answer: There is an 90%

chance that two of you share a birthday!

What’s this got to do with Stats?

• Remember that our assessment of statistical significance has to do with the judgment about whether or not an event happened because of some treatment or by chance.

• Probability gives us the tools to calculate the “by chance” part of this.

Defining Probability

• We define probability by comparing an outcome or set of outcomes with the set of all possible outcomes for an event.

• This will lead us to an “intuitive” definition of probability

Examples…• A coin toss:

– Two possible outcomes H or T– Probability for H is 1 of the 2 or ½ = 0.5 = 50%

• You win the “Stats 300 Lottery”– 39 possible outcomes– Only 1 of you! Probability is 1/39 = 2.5%

• Odds of a full-house in Poker– There are 2,598,960 possible poker hands– There are 3,744 ways to get a full house or

3744/ 2,598,960 = 0.024% (1 in 4165 hands!)

Independent Events

• When events are independent – the outcome (or probability) of the one does not change the probability of the other.

• Example:– You flip a coin and get heads – what is the

probability that you heads on the next flip?– NOTE – this is not the same as asking what is

the probability of flipping two heads in succession

Four Possible Outcomes

Probability of HH is(1/2)(1/2) = 1/4

Probability Rules (for events)…

• A probability of 0 means an event never happens

• A probability of 1 means an event always happens

• Probability P is a number always between 0 and 1

Probability Rules (for events)…

• If the probability of an event A is P(A) then the probability that the event does not occur is 1-P(A)

• This is also called the compliment of A and is denoted AC

• Example: what is the probability of not rolling a 6 when using an honest die?Solution: P6 = 1/6, PC

6 = 1 - 1/6 = 5/6

Probability Rules (in pictures)…

• If events A and B are completely independent of each other (disjoint) then the probability of A or B happening is just:

( ) ( ) ( )P A B P A P B

Sample Questions…

• What is the probability of flipping 5 successive heads?

• What is the probability of flipping 3 heads in 5 tries?

• From your text: 4.8, 4.13,4.14

Probability Rules (in pictures)…

• If events A and B are independent of each other (but not disjoint) then the probability of A and B happening is just:

( ) ( ) ( )P A B P A P B

In conclusion…

• Make sure you know what is meant by intuitive probability and why we express this as a number between 0 and 1

• Review the rules on page 298

• Try 4.11, 4.17, 4.22