# CS 361: Probability & Statistics

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1 January 31, 2018 CS 361: Probability & Statistics Probability

2 Probability theory

3 Probability Reasoning about uncertain situations with formal models Allows us to compute probabilities Experiments will be our data generating process

4 Outcomes If we toss a fair coin a bunch of times we expect about the same number of heads and tails If we roll a die, we don t expect to see one number more often than any other We can formally state the set of outcomes (heads, tails) we expect from an experiment (flipping the coin)

5 Outcomes Tossing a fair coin once: {H, T} Tossing a die: {1, 2, 3, 4, 5, 6} Tossing two coins: {HH, HT, TH, TT}

6 Sample space The sample space is the set of all possible outcomes of an experiment, written

7 Example Three playing cards: King (K), Queen (Q), Knight (N). One is turned over randomly What is the sample space? {K, Q, N}

8 Example Suppose we flip a card, turn it back over, rearrange the cards and flip another. What is the sample space? {KK, KQ, KN, QQ, QK, QN, NN, NK, NQ}

9 Example A couple decides to have children until they have both a boy and a girl or until they have three children What is the sample space? {BG, GB, BBG, GGB, BBB, GGG}

10 Example: Monty Hall There are three doors: door #1, door #2, door #3. Behind two of them are goats, behind one is a car. The goats are indistinguishable If we open the doors and note what we observe the sample space is {CGG, GCG, GGC}

11 Monty hall #2 Consider the Monty Hall scenario with distinguishable goats, one male and one female {CFM, CMF, FCM, FMC, MCF, MCF} Notice how there are more outcomes here

12 More family planning A couple decides to have children They decide to have children until a girl and then a boy is born What is the sample space here? The set of all strings that end in GB and contain no other GBs As a regular expression: B*G+B Somewhere between two and infinite children

13 Sample spaces Each run of an experiment has exactly one outcome The set of all possible outcomes is the sample space We will need to think of sample spaces to think rigorously about probability Sample spaces can be finite or infinite

14 Probability We might like to think of how often we will see each particular outcome A if we repeat an experiment over and over If our experiment is flipping a coin and we repeat it a large number of times We probably expect the relative frequencies of heads and tails to be non-negative and we want the frequency of either outcome to be at most 1 Finally we want the relative frequencies of heads and tails to add up to 1

15 Probability More generally, for an experiment with sample space that we want each outcome A to satisfy intuition tells us probability of outcome A And we also want the following

16 Example If we have a biased coin where P(H) = 1/3 and P(T) = 2/3 and we toss it three million times, how many times will we expect to see heads? We will see close to a million heads and 2 million tails

17 Example We often look at experiments where each outcome is equally likely In the example earlier with 3 cards, a King, Queen, and Knight where we turn one over at random. If each is equally likely, what probability should we assign to each card?

18 Example Recall the Monty Hall setup: 3 doors, 2 with a goat, and one with a car If we open the first door, what is the probability that we see a goat? What is the probability we see the car? P(car) = 1/3, P(goat) = 2/3

19 Example Recall the Monty Hall setup with distinguishable goats What is the probability we find a female goat behind door #1? P(female goat) = 1/3

20 Events Our formalization of probability will be about rules to assign probability to sets of outcomes For example, we might flip a coin three times Our sample space, the set of all outcomes, is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Getting two tails might be something we are interested int he probability of and is the set of outcomes {HTT, THT, TTH} We give sets of outcomes a special name, events, and the theory we develop will concern the probabilities of events

21 Events Remember: events are sets of outcomes The set of all outcomes is therefore an event and has probability 1 Likewise the empty set is an event (the set of no outcomes) and has probability 0

22 Events Since events are sets, we want it to be the case that if A and B are events things like,,, and are also events Example: Roll a die, let A={1,2,5} and B={2,4,6} What are,,, and in this case?

23 Probability and events: axioms Probabilities of events are non-negative: Every experiment has an outcome: The probability of disjoint events is additive: if for all i and j we have then Any function P that maps sets to real numbers and satisfies these is called a probability function or probability

24 Some properties These properties follow from the simple axioms on the last slide

25 Probability and size It can be helpful to think of the probability of an event as measuring its size We can then use diagrams to reason about certain properties. Which properties are illustrated below?

26 Counting and probability In many cases, we will use counting in order to compute probabilities If we are interested in an event writing its probability as follows, a set of outcomes, we might think of Thus, if each outcome in has the same probability then we have

27 Example Roll two fair dice and consider the total: what is the probability of getting an odd number? There are 36 outcomes in the sample space and each of them are equally likely giving each a probability of 1/36 18 of the 36 outcomes result in an odd number So the probability is 18/36=1/2

28 Example If we shuffle a pack of cards and draw one at random, what is the probability that it is a red 10? There are 52 cards, each an equally likely outcome in the sample space Our event of interest is the set containing the two outcomes where a red 10 is drawn The probability is 2/52=1/26

29 Example Roll two dice, add the two numbers, what in the probability the result is divisible by 5? There are 36 equally likely outcomes as before There are 4 ways to get a 5 with two dice and 3 ways to get a 10, so the probability is 7/36

30 Example We stop three random people on the street and ask them on what day of the week they were born. What is the probability that they were born on days of the week that are in succession? E.g. Monday-Tuesday-Wednesday or Saturday-Sunday-Monday The sample space has 7^3 outcomes each equally likely Our event has 7 possible outcomes So our probability is 7/7^3 = 1/49

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