What is Probability?

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 Theory Blaise Pascal ( ) Pierre Fermat ( ) 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 A C Example: what is the probability of not rolling a 6 when using an honest die? Solution: P 6 = 1/6, P C 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:

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:

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