Transcript lecture10
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 (1623-1662) The Gambler’s Dispute… Pierre Fermat (1601-1665) The gambler’s dispute (1654) • This famous dispute led to the formal development of the mathematical theory of probability "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. 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 Probability of HH is (1/2)(1/2) = 1/4 Four Possible Outcomes Probability Rules (for events)… • A probability of 0 means an event never happens • A probability of 1 means an event always happens P • Probability 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, PC6 = 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 • 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 pictures)… 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