2 What 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.
3 Probability 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.
4 What 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
5 Estimated 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?
6 Introduction 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.
7 Useful 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
8 c. 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
9 Combining 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=
10 Combining 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?
11 Combining 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
12 ProbabilityThrowing 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
13 Probability 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.
14 ProbabilityThe probability that they will win can only be estimated, using other information, mainly past performance of the two teams.