Chapter 22 H- Venn Diagrams Roll a die U = {1, 2, 3, 4, 5, 6} We want a number less than Event “A”

U = {1, 2, 3, 4, 5, 6} A = A

This Venn diagram shows the set U of all children in a class. The event E shows all those students with Blue Eyes. Determine the probability that a randomly selected child a) Has Blue Eyesb) Doesn’t have Blue Eyes E E’ (Not Blue Eyes)

EE’ = (Not Blue Eyes) 8 15 n(U) = 23 n(E) = 8

In an IB Math SL class with 30 students: 19 Study Physics 17 Study Chemistry 15 Study BOTH Physics and Chemistry Use a Venn Diagram and determine the probability that a randomly selected student studies: a)Both Subjects b)At least one of the subjects c) Physics, but NOT Chemistry d)Exactly 1 of these subjects e)Neither subject

P = The Event of studying Physics C = The Event of studying Chemistry a + b = 19 (Studying Physics) b + c = 17 (Studying Chemistry) b = 15 (Study Both) d = Students who don ’ t study either Physics nor Chem. P C U a b c d b = 15 a + b + c + d = 30 a = 4, b = 15, c = 2 d = 9

P C U a) P(Both) b) P (At least 1 subject) c) P(P but not C) d) P(Studies exactly 1) e) P(Neither) U = 30

Suppose a six-sided die is rolled. The event that the die would land on an even number would be E = {2, 4, 6} The event that the die would land on a prime number would be P = {2, 3, 5} What would be the event E and P happening? E and P = {2} This is an example of the intersection of two events.

The intersection of A and B - consists of all outcomes that are in both of the events This symbol means “intersection”

Let’s revisit rolling a die and getting an even or a prime number... E and P = {2} To represent this with a Venn Diagram: E and P would be ONLY the middle part that the circles have in common

Union The union of any collection of events is the event that at least one of the collection occurs.

Suppose a six-sided die is rolled. The event that the die would land on an even number would be E = {2, 4, 6} The event that the die would land on a prime number would be P = {2, 3, 5} What would be the event E or P happening? E or P = {2, 3, 4, 5, 6} This is an example of the union of two events.

The union of A or B - consists of all outcomes that are in at least one of the two events, that is, in A or in B or in both. This symbol means “union” Consider a marriage or union of two people – when two people marry, what do they do with their possessions ? The bride takes all her stuff & the groom takes all his stuff & they put it together! And live happily ever after! This is similar to the union of A and B. All of A and all of B are put together!

Let’s revisit rolling a die and getting an even or a prime number... E or P = {2, 3, 4, 5, 6} Another way to show this is with a Venn Diagram. Even number Prime number E or P would be any number in either circle. Why is the number 1 outside the circles?

Unions: , or Intersections: , and A  A 

General Rule for Addition For any two events A and B, AB Since the intersection is added in twice, we subtract out the intersection.

The Addition Law of Probability OR

0.7 = P(B) – 0.3 P(B) = 0.4 U A B a b 0.3 a = 0.6 & a + b = 0.7 a = 0.3 a + b = b = 0.4 b = 0.1 P(B) = b = 0.4

Mutually Exclusive or Disjoint Sets A B S Sets have nothing in common

Mutually Exclusive or Disjoint Sets If A and B are mutually exclusive events, then The additional law, then becomes:

Of the 31 Freshman in my 9 th grade class, 7 were born in Transylvania (T) 5 were born ON the North Pole. (N) a)Are T and N mutually exclusive events? YES. You can’t be born in two place at once.

If a student in the class was chosen at random, find the probability that he or she was born in i. Transylvania ii. North Pole iii. T or N

Homework Page 592 (Odds)

Conditional Probability Conditional Probability contains a condition that may limit the sample space for an event. You can write a conditional probability using the notation - This reads “the probability of event B, given event A”

The table shows the results of a class survey. Find P(own a pet | female) The condition female limits the sample space to 14 possible outcomes. Of the 14 females, 8 own a pet. Therefore, P(own a pet | female) equals. yesno female86 male57 Do you own a pet? 14 females; 13 males 8 14

The table shows the results of a class survey. Find P(Ate Taco Bell| male) Conditional Probability The condition male limits the sample space to 15 possible outcomes. Of the 15 males, 7 Ate Taco Bell. Therefore, P(Ate Taco Bell | male) yesno female76 male78 Did you eat Taco Bell last night? 13 females; 15 males 7 15

Conditional Probability Formula For any two events A and B from a sample space with P(A) does not equal zero

In an IB class of 25 students, 14 like Tofu Milk Shakes with Lemongrass and 16 like deep fried grubs. One student likes neither and 6 students like both. What is the probability that the student: A) Likes Tofu ShakesB) Likes Tofu shakes given that they like fried grubs. T G U A) P(TOFU) = B) P(Tofu|Grubs) =

You Try In a class of 40 students, 34 Like Lady Gaga, 22 like bananas, and 2 dislike both. Find the probability that the student: A) Likes both Lady Gaga and a banana B) Likes at least Lady Gaga and a banana C) Like a Banana given they like Lady Gaga D) Dislikes Lady Gaga given they like bananas

In a class of 40 students, 34 Like Lady Gaga, 22 like bananas, and 2 dislike both. G = students who like Lady Gaga B = students who like bananas. G B U abc 2 Given: a + b = 34 b + c = 22 a + b + c = 38 c = 38 – 34 = 4 b = 18 a = 16

G B U A) P(Both) = B) P(Likes at least 1) = C) P(G|B) = D) P(B’|G) =

Homework Page 598 (Evens) Due at end of class.