1 Probability Part 1 Definitions Event Probability Union

1 Probability Part 1 – Definitions * Event * Probability * Union * Intersection * Complement Part 2 – Rules Probability

2 Definitions - Event * An event is a specific collection of possible outcomes. * For example, the event ‘A’ could be the event that one Head occurs in two tosses of a coin. HH HT TH A Probability TT

3 Definitions - Probability The probability of an event is the ratio of the number of outcomes matching the event description to the number of possible outcomes. P(1 head in 2 tosses) = 2/4 =. 5 Probabilities are ALWAYS between 0 and 1. Probability

4 Definitions - Union The union of two events A and B is the event that occurs if either A or B or both occur on a single measurement. * For example, suppose: * A = a car has two doors * B = a car is red * Then, if we randomly select a car from a large parking lot, P(A U B) is the probability that the car is either a two -door model or red or both two-door and red. Probability

5 Definitions – Union Define: C = being female D = being nineteen years old What is P(C U D) for a randomly selected person from among the set of people currently in this room? Probability

6 Definitions – Union # of women = ____ plus # of 19 year olds = ____ minus # of 19 year old women = ____ equals # either 19 or a woman or both = ____ Probability

7 Definition – Union Total # of people in this room = _____ Therefore, P(C U D) = _____ = Probability

8 Definitions – Intersection The intersection of two events A and B is the event that occurs if and only if both A and B occur on a single measurement. * Suppose we randomly select a person from the set of people currently in this room. With A and B defined as before, the probability that a randomly-selected person is a 19 year old woman is: P(A ∩ B) = Probability

9 Definitions – Complement The complement of an event A is the event that A does not occur. A’ HH TT HT TH A Probability

10 Definitions – Complement P(A’) = 1 – P(A) + P(A’) = 1 The second equation says, “Either A happens or A doesn’t happen. There are no other possibilities. ” That may seem obvious, but keep it in mind on exams. Probability

11 Part 2 – Rules * Additive Rule * Mutually Exclusive Events * Conditional Probability * Multiplicative Rule * Independence Probability

12 Rules – Additive Rule P(A U B) = P(A) + P(B) – P(A ∩ B) * Recall 19 year old women example. * When you add # of 19 year olds to # of women, you count the 19 year old women in both groups – so you count them twice. Subtract that number once as a correction. Probability

13 Rules – Mutually Exclusive Events Two events are mutually exclusive when P(A ∩ B) = 0 and P(A U B) = P(A) + P(B) Be careful! Note that the first equation uses ∩ while the second one uses U. Probability

14 Rules – Conditional Probability When you have information that reduces the set of possible outcomes, you work with new probabilities that are conditional on that new information. *Remember that the # of possible outcomes is the denominator of the ratio that gives probability of an event. *The probability changes on the basis of new information because the numerator stays the same but the denominator decreases. Probability

15 Rules – Conditional Probability Example: Suppose before class I picked a card at random from a standard deck of cards. What is P(E) if: E = Card I picked is a Club? * Note that this is a question about the ordinary (nonconditional) probability. Probability

16 Rules – Conditional Probability P(E) = 13/52 =. 25 (Do you see why? ) Now, suppose I tell you that the card I picked is black. What is the conditional probability that the card is a club given that it is black? P(E │black) = 13/26 =. 5 * Only 26 cards in a normal deck are black. Probability

17 Rules – Conditional Probability We write: P(A │B) = P(A ∩ B) P(B) Probability

18 A B A∩B P(A │B) = P(A ∩ B) P(B) Probability

19 Rules – Multiplicative Probability P(A ∩ B) = P(A) * P(B│A) P(A ∩ B) = P(B) * P(A│B) Probability

20 Rules – Multiplicative Probability To see why, begin with the conditional probability formula, and multiply both sides by either P(A) or P(B): P(A │B) = P(A ∩ B) P(B │A) = P(A ∩ B) P(A) Probability

21 Rules - Independence Events A and B are independent if the occurrence of one does not alter the probability of the other. P(A│B) = P(A) P(B│A) = P(B) Probability

22 Rules – Independence We can now re-write the multiplicative rule for the special case of independent events: P(A ∩ B) = P(A) * P(B) * This is because P(B│A) = P(B) for independent events. Probability

23 Probability – Examples 60% of Western students are female. 60% of female students have a B average or better. 80% of male students have less than a B average. a. What is the probability that a randomly selected student will have less than a B average? Probability

24 A = Being female B = Having a B average or better B . 6 x. 6 =. 36 . 6 A. 6 . 4. 2 B’ B . 6 x. 4 =. 24. 4 x. 2 =. 08 . 4 A’. 8 B’ Probability . 4 x. 8 =. 32

25 Probability – Examples a. What is the probability that a randomly selected student will have less than a B average? P(B’) = P(B’ ∩ A) + P(B’ ∩ A’) * Either we get (B’ and A) or we get (B’ and A’). * That is, either our randomly selected student who has less than a B average is a female or he is a male. Probability

26 Probability – Examples P(B’) = P(B’ ∩ A) + P(B’ ∩ A’) By Multiplicative Rule: P(B’) = P(B’│A)*P(A) + P(B’│A’)*P(A’) = (. 4*. 6) + (. 8*. 4) =. 56 Probability

27 Reminder – Multiplicative Probability P(A ∩ B) = P(A) * P(B│A) P(A ∩ B) = P(B) * P(A│B) Probability

28 Probability – Examples b. If we randomly select a student at Western and note that this student has a B or better average, what is the probability that the student is male? Probability

29 A = Being female B = Having a B average or better B . 6 x. 6 =. 36 . 6 A. 6 . 4. 2 B’ B . 6 x. 4 =. 24. 4 x. 2 =. 08 . 4 A’. 8 B’ Probability . 4 x. 8 =. 32

30 Probability – Examples We know that overall, 40% of students (A’) are male. But among the students who get a B or better average, what proportion are male? The probability of getting a B or better average is. 44 (from. 36 for women and. 08 for men). Thus, P(A’│B) = P(A’ ∩ B) P(B) =. 08. 44 Probability =. 1818

31 Probability – Examples You’re on a game show. You’re given a choice of 3 doors you can open. Behind one door is a car. Behind each of the other two doors is a goat. You win what is behind the door you open. You pick a door, but don’t get to open it yet. The host opens another door, behind which is a goat. The host then says to you, “Do you want to stick with the door you chose or switch to the other remaining door? ” Is it to your advantage to switch? Probability

32 Probability – Examples Yes – you should switch to the other door. The door you originally chose has a 1/3 rd chance of winning the car. The other remaining door has a 2/3 rd chance of winning the car. Probability

33 Probability – Examples Suppose you originally pick Door #1. The door the host opens is indicated by the boldface type below. What you win if you switch is underlined. Door #1 Door #2 Door #3 Goat Car Goat Probability

34 Probability - Examples When you picked Door #1, there was a 2/3 rd probability that the car was behind one of the other doors. * That is still true after the host opens one of those other doors. * But since the host knows where the car is, he opens a door that has a goat behind it. So now the 2/3 rd probability is all associated with the one remaining door. * The critical point is that the host has information – so we are dealing with the conditional probability of the car being behind Door #3 GIVEN THAT the host opened Door #2 (after you picked Door #1). Probability