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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