2What is it?For a random phenomenon, individual outcomes are not certain, but there is a regular distribution of outcomes in the long runThe probability of an outcome is its long-term relative frequency.
3Probability vs. inference If a coin is tossed 100 times, you would expect to see 50 headsHowever, there is some (non-zero) probability that it only comes up heads 25 timesIf there were 100 flips, and it only came up heads 25 times, you might infer that it wasn’t a fair coin.
4What are the possible outcomes? Want to make a list of possible outcomes, then find probability for each outcomeSample space is the set of all possible outcomesEvents are specific outcomes or set of outcomes in the sample space
5Estimated Probability Probability ScaleYou can draw a scale from Impossible to Certain:Estimated ProbabilityImpossibleUnlikelyEven ChanceLikelyCertainWhat is the probability of scoring less than 5 when rolling a fair 6-sided die?
6Introduction to Probability Rather than using a descriptive scale, we use a numerical scaleProbability is a numerical measure of the likelihood that an event will occur.Probability values are always assigned on a scale from 0 to 1.A probability near 0 indicates an event is very unlikely to occur.A probability near 1 indicates an event is almost certain to occur.
7Useful facts about probability Probability cannot be less than 0 or greater than 1.All possible outcomes together must have probability 1.Probability of an event occurring is 1 minus probability that it does not occur.If 2 events have no outcomes in common, probability that one or the other occurs is sum of their individual probabilities
8c. rolling a number less than 7 Example: You roll a six-sided die whose sides are numbered from 1 through 6. Find the probability ofa. rolling a 4b. rolling an odd numberc. rolling a number less than 7= number of ways to roll a 4number of ways to roll the die= 16= number of ways to roll an odd numbernumber of ways to roll the die= 36= 12= number of ways to roll less than 7number of ways to roll the die= 66= 1
9Combining Probabilities What is the probability of scoring either a 2 or a 6 from a single roll of a fair die?Number of ways an event can happen Total number of possible outcomes26=13=16161616So p(1 or 5) =Or p(1) =And p(5) =+26=13=1 + 16=
10Combining probabilities What happens if we need to calculate the probability of one event occurring and another event occurring?What is the probability of rolling a 4 with a die and tossing a head with a coin?How many outcomes are there?
11Combining probabilities 123456H1H2H3H4H5H6T1T2T3T4T5T6HT12 possible outcomes.Only one is H plus a 4112So p(4 and head) =16Or: The probability of rolling a 4 is12The probability of throwing a head is12112161216So p(4 and head) =of=of the time
12ProbabilityThrowing dice, tossing coins, picking cards - in each case each outcome is equally likelyIt is not always the case that all possible outcomes of a trial are equally likely.Sometimes we cannot calculate the probability, but can only estimate it
13Probability Will it rain tomorrow? Will Team A win over Team B? Each outcome is not equally likely.For one team playing a game against another, there are three possible outcomes: Win, Lose, DrawProbability of winning is not one third.
14ProbabilityThe probability that they will win can only be estimated, using other information, mainly past performance of the two teams.