13.4 – Compound Probability Chapter 13 – Probability 13.4 – Compound Probability

Compound Events A compound event is an event that is made up of 2 or more events. 2 Types: Independent Events – if the occurrence of one event does not affect how the other event occurs Dependent Events – if the occurrence of one event does affect how the other event occurs.

does not Independent Events Events where the occurrence of one of the events ______ _____ affect the occurrence of the other event. does not A and B are independent if and only if: P(A ∩ B) = P(A) × P(B) P(A and B) = P(A) × P(B) “and” → multiplication P(A and B and C) = P(A) × P(B) × P(C)

Independent Events If one student in the class was born on June 1st can another student also be born on June 1st? If you roll a die and get a 6, can you flip a coin and get tails?

Independent Events P(A and B) = P(A) × P(B) P(A ∩ B) = P(A) × P(B) Practice: A coin and a die are tossed simultaneously. Determine the probability of getting heads and a 3. P(A and B) = P(A) × P(B) P(A ∩ B) = P(A) × P(B) P(H and 3) = P(H) × P(3) = 𝟏 𝟐 x 𝟏 𝟔 = 𝟏 𝟏𝟐

P(B and B) = P(B) × P(B) = 𝟗 𝟏𝟓 x 𝟗 𝟏𝟓 = 𝟖𝟏 𝟐𝟐𝟓 Independent Events Practice: There are 9 brown boxes and 6 red boxes on a shelf. Anna chooses a box and replaces it. Brian does the same thing. What is the probability that Anna and Brian choose a brown box? P(B and B) = P(B) × P(B) = 𝟗 𝟏𝟓 x 𝟗 𝟏𝟓 = 𝟖𝟏 𝟐𝟐𝟓

Independent Events Practice P (A) = 2 9 P (B) = 3 5 and P(A and B) = 2 15 Are A and B independent events? Explain your answer. A and B are independent if P(A and B) = P(A) × P(B). Independent events 𝟐 𝟗 x 𝟑 𝟓 = 𝟔 𝟒𝟓 = 𝟐 𝟏𝟓 , Thus they are independent

Mutually Exclusive Events A bag of candy contains 12 red candies and 8 yellow candies. Can you select one candy that is both red and yellow?

Mutually Exclusive Events Two events, A and B are mutually exclusive if whenever A occurs it is impossible for B to occur and vice-versa. Events such as A and A’ must be mutually exclusive. If events A and B are mutually exclusive, then A ∩ B = ∅ Events A and B are mutually exclusive if and only if P(A and B) = 0 or P(A ∩ B) = 0

Mutually Exclusive Events Please Note: If events are mutually exclusive, then the probability of either one event or the other event occurring is given by: P(either A or B) = P(A) + P(B)

Mutually Exclusive Events Practice: Of the 31 people on a bus tour, 7 were born in Scotland and 5 were born in Wales. Are these events mutually exclusive? If a person is chosen at random, find the probability that he or she was born in: Scotland Wales Scotland or Wales Yes, what is the probability of being born in Scotland AND Wales? Zero! P(S) = 𝟕 𝟑𝟏 Mutually exclusive events P(W) = 𝟓 𝟑𝟏 P(S or W) = P(S) + P(W) = 𝟕 𝟑𝟏 x 𝟓 𝟑𝟏 = 𝟏𝟐 𝟑𝟏

P(A or B) = P(A) + P(B) – P(A and B) P(A ∪ B) = P(A) + P(B) – P(A ∩ B) Overlapping Events Overlapping Events or Combined Events ~Have outcomes that are in common~ Like the example of drawing a number 1 through 20 that is a multiple of 3 or 5. Drawing the number 15 would fit both of these outcomes For overlapping events: P(A or B) = P(A) + P(B) – P(A and B) P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Overlapping Events Solution: Example: 50 people are members at a gym. 28 of those people play tennis. 14 of those people play basketball and 7 play both tennis and basketball. What is the probability that a randomly selected person plays tennis or basketball? Solution: P(T) = 28 50 , P(B) = 14 50 , P(T ∩ B) = 7 50 28 50 + 14 50 - 7 50 = 35 50 = 7 10

Overlapping Events Practice 100 people were surveyed: 72 people have had a beach holiday 16 have had a skiing holiday 12 have had both What is the probability that a person chosen has had a beach holiday or a ski holiday? Solution: P(B) = 72 100 , P(S) = 16 100 , P(T and B) = 12 100 72 100 + 16 100 - 12 100 = 74 100 = 0.74

If P(A) = 0.6 and P(A or B) = 0.7 and P(A and B) = 0.3, find P(B). Overlapping Events Practice If P(A) = 0.6 and P(A or B) = 0.7 and P(A and B) = 0.3, find P(B). Solution: P(A or B) = P(A) + P(B) – P(A and B) P(B) = P(A or B) – P(A) + P(A and B) P(B) = 0.7 - 0.6 + 0.3 = 0.4 Combined events (inclusive events)

There are 9 brown boxes and 6 red boxes on a shelf. Dependent Events There are 9 brown boxes and 6 red boxes on a shelf. What if Anna choose the box and did not replace it? Then Brian’s event of choosing a box becomes dependent. If Anna chooses red, P(Brian chooses brown) = If Anna chooses brown, P(Brian chooses brown) = P(Anna then Brian choose brown)

Dependent Events Events where the occurrence of one of the events ______ affect the occurrence of the other event. does P(A then B) = P(A) × P(B given that A has occurred)

Dependent Events P(A then B) = P(A) × P(B given that A has occurred) Practice: A box contains 4 red and 2 yellow tickets. Two tickets are randomly selected one by one from the box, without replacement. Find the probability that: (a) both are red (b) the first is red and the second is yellow. P(A then B) = P(A) × P(B given that A has occurred) P(2R) = 𝟒 𝟔 x 𝟑 𝟓 = 𝟏𝟐 𝟑𝟎 = 𝟐 𝟓 P(RY) = 𝟒 𝟔 x 𝟐 𝟓 = 𝟖 𝟑𝟎 = 𝟒 𝟏𝟓

Dependent Events Prime Numbers = {2,3,5,7,11,13,17,19} Practice: A hat contains tickets with numbers 1, 2, 3, … , 19, 20 printed on them. If 3 tickets are draw from the hat, without replacement, determine the probability that all are prime numbers. Prime Numbers = {2,3,5,7,11,13,17,19} P(3P) = 𝟖 𝟐𝟎 x 𝟕 𝟏𝟗 x 𝟔 𝟏𝟖 = 𝟑𝟑𝟔 𝟔𝟖𝟒𝟎 = 0.049

Laws of Probability Type Definition Formula Mutually Exclusive Events events that cannot happen at the same time P(A and B) = 0 / P(A ∩ B) = 0 P(A or B) = P(A) + P(B) P(A  B) = P(A) + P(B) Overlapping Events events that can happen at the same time P(A or B) = P(A) + P(B) – P(A and B) P(AB) = P(A) + P(B) – P(A∩B) Independent Events occurrence of one event does NOT affect the occurrence of the other P(A and B) = P(A) • P(B) P(A ∩ B) = P(A) • P(B) Dependent Events occurrence of one DOES affect the occurrence of the other P(A then B) = P(A) • P(B)

A sample space is the set of all possible outcomes of an experiment. 4 ways to create a Sample Set: List the outcomes {you must use curly brackets} Table of outcomes Draw a 2 Dimensional Grid Draw a Tree Diagram

Sample Space List the outcomes for: Practice a) Flipping a coin b) Rolling a single die c) Flipping two coins d) Rolling two dice

Sample Space Practice Use a 2D grid to illustrate the sample space for tossing a coin and rolling a die simultaneously. Find the probability of: Rolling a Die 1 2 3 4 5 6 Coin Toss H H1 H2 H3 H4 H5 H6 T T1 T2 T3 T4 T5 T6 a) tossing heads on the coin b) getting tails and a 5 c) getting tails or a 5

Sample Space Practice Two square spinners, each with 1, 2, 3, and 4 on their edges, are twirled simultaneously. Draw a 2D grid of the possible outcomes. Find the probability of: Spin 2 1 2 3 4 Spin 1 (1,1) (1,2) (1,3) (1,4) (2,1) (2,2) (2,3) (2,4) (3,1) (3,2) (3,3) (3,4) (4,1) (4,2) (4,3) (4,4) a) getting a 3 with each spinner b) getting a 3 and a 1 c) getting an even result for each spinner

Sample Space Practice: A red and blue dice are rolled together. Calculate the probability that: (a) The total score is 7 (b) The same number comes up on both dice (c) The difference between the scores is 1 (d) The score on the red dice is less than the score on the blue dice (e) The total score is a prime number Roll 2 1 2 3 4 5 6 Roll 1 (1,1) 2 (1,2) 3 (1,3) 4 (1,4) 5 (1,5) 6 (1,6) 7 (2,1) 3 (2,2) 4 (2,3) 5 (2,4) 6 (2,5) 7 (2,6) 8 (3,1) 4 (3,2) 5 (3,3) 6 (3,4) 7 (3,5) 8 (3,6) 9 (4,1) 5 (4,2) 6 (4,3) 7 (4,4) 8 (4,5) 9 (4,6) 10 (5,1) 6 (5,2) 7 (5,3) 8 (5,4) 9 (5,5) 10 (5,6) 11 (6,1) 7 (6,2) 8 (6,3) 9 (6,4) 10 (6,5) 11 (6,6) 12 Example 9 from page 503 of 2nd edition

Sample Space Practice Draw a table of outcomes to display the possible results when two dice are rolled and the scores are summed. Determine the probability that the sum of the dice is 7. Roll 2 1 2 3 4 5 6 Roll 1 7 8 9 10 11 12 Example 9 from page 503 of 2nd edition