Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Aim: How do we determine the probability of compound events? Do Now: What is the probability of flipping a regular coin and getting a head? P(H) = Inside a baggie are 1 red, 1 pink, 1 blue and 1 green chip. What is the probability of reaching into the baggie and picking a green chip? P(G) = What is the probability of reaching into the baggie and picking a green chip and flipping a coin and getting heads? P(G and H) = Describe how the first two problems are different from the third. 1/2 1/4 1/2 1/4 = 1/8

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. pink H pink T red H red T blue H blue T green H green T Sample Space 8 outcomes Flip Coin Construct a tree diagram to show all possible outcomes for the 3rd part of the Do Now problem. Find P(pink, H) Pick Chip pink red blue green HTHT HTHT HTHT HTHT Event A Event B P(pink) = 1/4 P(H) = 1/2 P(pink, H) = 1/4 1/2 = 1/8 P(p H) = 1 8 IndependentIndependent

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Compound Event Compound Event - two or more activities Ex. Rolling a pair of dice What is the probability of rolling a pair of dice and getting a total of four? Die Die 2 P(4) = 3/36 = 1/12

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Independent Events Independent Events – two events are independent if the occurrence of either of them has not effect on the probability of the other. Mutually exclusive – two events A & B are mutually exclusive if they can not occur at the same time. That is, A and B are mutually exclusive when A  B =  Independent events can occur at the same or different times, and have no effect on each other. Ex. mutually exclusive? rolling a 2 and a 3 on a die rolling an even number or a multiple of 3 on a die Yes No

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Probability of Two Independent Events The probability of two independent events can be found by multiplying the probability of the first event by the probability of the second event. P(A and B) = P(A) · P(B) Ex: A die is tossed and a spinner is spun. What’s the probability of throwing a 5 and spinning red? P(5 and R)? P(5 and Red) = Faster than drawing a tree diagram!! P()5 1 6  Pred()  1 4 Event A Event B AND Probabilities with Independent Events If A and B are independent events, then P(A and B) = P(A) · P(B)

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Model Problem A special family has had nine girls in a row. Find the probability of this occurrence. Having a girl is an independent event with P(1 girl) = 1/2 P(A and B) = P(A) · P(B) Probability of two Independent Events extends to multiple independent events

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Model Problem If the probability that South Florida will be hit by a hurricane in any single year is 5/19 a)What is the probability that S. Florida will be hit by a hurricane in three consecutive years? b)What is the probability that S. Florida will not be hit by a hurricane in the next ten years? The probability of event (A) plus the probability of "not A” or ~A, equals 1: P(A) + P(~A) = 1; P(A) = 1 – P(~A); P(~A) = 1 – P(A)

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Not So Independent! There are 4 red, 3 pink, 2 green and 1 blue chips in a bag. What is P(pink)? 3/10 What is the probability of picking a pink and then reaching in and picking a second pink w/o replacing the first one picked? 6/90 or 1/15 is the probability of picking a pink chip and then picking a second pink chip. BUT ONLY IF THE FIRST PINK CHIP WAS NOT RETURNED TO THE BAG. The selection of the second event was affected by the selection of the first. Dependent Events

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Dependent Events Two events are dependent events if the occurrence of one of them has an effect on the probability of the other. AND Probabilities with Dependent Events If A and B are dependent events, then P(A and B) = P(A) · P(B given that A has occurred) extends to multiple dependent events You are dealt three cards from a 52-card deck. Find the probability of getting 3 hearts. P(1 st heart) = 13/52 P(2 nd heart) = 12/51 P(3 rd heart) = 11/50 P(hearts) = 13/52 · 12/51 · 11/50 = 1716/ 

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Model Problem Three people are randomly selected, one person at a time, from 5 freshman, two sophomores, and four juniors. Find the probability that the first two people selected are freshmen and the third is a junior. P(1 st selection is freshman)= 5/11 P(2 nd selection is freshman)= 4/10 P(3 rd selection is junior)= 4/9 P(F, F, J) = 5/11 · 4/10 · 4/9 = 8/99

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Model Problem P(pink) Event A 3/10 Event B P(pink) 2/9 Dependent Events P(pink, pink) = Find the probability of choosing two pink chips without replacement. 3/10 2/9 = 6/90 or 1/15 Counting Principle w/Probabilities

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Model Problem P(blue) Event A 1/10 Event B P(red) 4/91/10 4/9 = 4/90 or 2/45 Find the probability of choosing blue and then a red chip without replacement. Dependent Events P(blue, red) = Counting Principle w/Probabilities

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Probability of Dependent Events 1. Calculate the probability of the first event. 2. Calculate the probability of the second event, etc.... but NOTE: The sample space for the probability of the subsequent event is reduced because of the previous events. 3. Multiply the the probabilities. Ex. A bag contains 3 marbles, 2 black and one white. Select one marble and then, without replacing it in the bag, select a second marble. What is the probability of selecting first a black and then a white marble? Key words - without replacement

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Model Problem From a deck of 10 cards (5 ten-point cards, 3 twenty-point cards, and 2 fifty-point cards), Ronnie can only pick 2 cards. In order to win the game, he must pick the 2 fifty-point cards. What is the probability that he will win?

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Model Problem From a deck of 10 cards Ronnie can only pick 2 cards. In order to win the game, he must pick the 2 fifty-point cards. What is the probability that he will win? Dependent P(50, 50) = 2/10 1/9 = 2/90 = 1/45 Counting Principle w/Probabilities Event A Event B P(50) = 2/10 P(50) = 1/9 50

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Model Problems Penny has 3 boxes, each containing 10 colored balls. The first box contains 1 red ball and 9 white balls, the second box contains 3 red balls and 7 white balls, and the third box contains 7 red balls and 3 white balls. Penny pulls 1 ball out of each box. Box 1Box 2Box 3 A. What is the probability that Penny pulled 3 red balls? P(r,r,r) = 1/10 3/10 7/10 = 21/1000 B. If Penny pulled 3 white balls and did not replace them, what is the probability that she will now pull 3 red balls? P(r,r,r) = 1/9 3/9 7/9 = 21/729

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Model Problems A sack contains red marbles and green marbles. If one marble is drawn at random, the probability that it is red is 3/4. Five red marbles are removed from the sack. Now, if one marble is drawn, the probability that it is red is 2/3. How many red and how many green marbles were in the sack at the start? x = original red marbles y = original number of green marbles x__ x + y 3 4 = 5 15 x - 5 x + y = 3x + 3y = 4x2x + 2y - 10 = 3x y = x2y + 5 = x 3y = 2y + 5 y = 5 3y = x = 15

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Conditional Probability of A and B Conditional Probability - The probability of event B, assuming that the event A has already occurred, is call the conditional probability of B, given A. This is denoted by P(B|A). # outcomes of B that are in the restricted sample space of A P(B|A) = # outcomes in the restricted sample space of A

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Model Problem Conditional probabilities are calculated based on common outcomes regarding the two events A and B; A  B. Find the probability of rolling a die and getting a number that is both odd and greater than 2. {3, 5} {1, 3, 5} {3, 4, 5, 6}

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Conditional Probability of A and B Region I Region II Region IV AB Reg. III U The intersection of sets A and B is region III. ‘and’ is the term used to describe intersection The intersection of sets A and B, denoted by A  B, is the set consisting of all elements common to A and B. A  B = {x|x  A AND x  B}   recall:

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Venn Diagrams Find the probability of rolling a die and getting a number that is both odd and greater than odd > 2 P(odd) = 3/6P(> 2) = 4/6 P(odd  > 2) = 2/6

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Model Problem A letter is randomly selected from the letters of the English alphabet. Find the probability of selecting a vowel, given that the outcome is a letter that precedes h. S = {a, b, c, d, e, f, g} sample space

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Model Problem The table below shows the differences in political ideology per 100 males and per 100 females in the 2000 US presidential election. Find the probability that the person a.is liberal, given that the person is female b.is male, given that the person is conservative. LiberalModerateConservative male female = 100

Aim: ‘And’ Probabilities & Independent Events Course: Math Lit. Model Problems A pair of dice are tossed. Find the probability that the sum on the two dice is 8, given that the sum is even. A pair of dice are tossed twice. Find the probability that both rolls give a sum of 8. Find the probability that 1 person selected is moderate, given college attendance. LiberalModerateConservative HS only73513 College101520