Chapter 10 PROBABILITY

Probability Terminology  Experiment: take a measurement Like flipping a coin  Outcome: one possible result of an experiment.  Sample space: a set of all possible outcomes  Event: any collection of possible outcomes.

Example 1  The experiment consists of rolling a six- sided die and recording the number.  List the sample space.  List one possible event.

Example 1, cont’d  The sample space has 6 possible outcomes and is {1, 2, 3, 4, 5, 6}.  One possible event is the event of getting an even number: {2, 4, 6}.

Example 2  The experiment is tossing a coin 3 times and recording the results in order.  What’s the sample space?  What are some possible events?

Example 2  The experiment is tossing a coin 3 times and recording the results in order.  The sample space is {HHH, HHT, HTH, HTT, TTT, TTH, THT, THH}  What are some possible events?  One possible event is tossing Heads first {HHH, HHT, HTH, HTT}

Probability, cont’d  Probability: a number from 0 to 1, and can be written as a fraction, decimal, or percent.  The greater the probability, the more likely an event is to occur.  An impossible event has probability 0.  A certain event has probability 1.

Experimental Probability  Experimental probability: how often an event occurs in a particular sequence of trials.

Example 5  An experiment consisted of tossing 2 coins 500 times and recording the results  If E is the event of getting a head on the first coin, find the experimental probability of E.

Theoretical Probability  Theoretical probability is the chance an event will occur based on the situation  like tossing a coin and knowing each side should come up half of the time.

Theoretical Probability, cont’d  If all outcomes are equally likely, the probability of event E is the number of outcomes in the event divided by the number of outcomes in the sample space.  The probability of event E is written P(E).

Example 6  An experiment consists of tossing 2 fair coins.  Find the theoretical probability of: a) The event E of getting a head on the first coin. b) The event of getting at least one head.

Example 6, cont’d  It helps to first find the sample space: {HH, HT, TT, TH}  The event E is {HH, HT} and the theoretical probability of E is:

Example 6, cont’d  The event of getting at least one head is E = {HH, HT, TH}.

Example 8  A jar contains four marbles: 1 red, 1 green, 1 yellow, and 1 white.

Example 8, cont’d  If we draw 2 marbles in a row, without replacing the first one, find the probability of: a) Event A: One of the marbles is red. b) Event B: The first marble is red or yellow. c) Event C: The marbles are the same color. d) Event D: The first marble is not white. e) Event E: Neither marble is blue.

Example 8, cont’d  The sample space has 12 outcomes: {RG, RY, RW, GR, GY, GW, YR, YG, YW, WR, WG, WY}. a) Event A: One of the marbles is red.  A = {RG, RY, RW, GR, YR, WR}. 

Example 8, cont’d b) Event B: The first marble is red or yellow.  B = {RG, RY, RW, YR, YG, YW}.  c) Event C: The marbles are the same color.  C = { }. 

Example 8, cont’d d) Event D: The first marble is not white.  D = {RG, RY, RW, GR, GY, GW, YR, YG, YW}.  e) Event E: Neither marble is blue.  E = all of S 

Union and Intersection  Union, A U B, means all outcomes that are in one, the other, or both events.  Intersection, A ∩ B, means outcomes that are in both events.

Mutually Exclusive Events  Mutually exclusive: Events that have no outcomes in common   If A and B are mutually exclusive events,

Example 9  A card is drawn from a standard deck of cards.  A is the event the card is a face card.  B is the event the card is a black 5.  Find P(A U B).

Example 9, cont’d  Event A has 12 outcomes, one for each of the 3 face cards in each of the 4 suits.  P(A) = 12/52.  Event B has 2 outcomes, because there are 2 black fives.  P(B) = 2/52.

Example 9, cont’d  Events A and B are mutually exclusive because a 5 cannot be a face card.  P(A U B) = 12/52 + 2/52 = 14/52 = 7/26.

Complement of an Event  Complement of an event E: outcomes in a sample space S, but not in the event E  The complement of E is written Ē.

Example 10  In a number matching game,  Carolyn chooses a whole number from 1 to 4.  Then Mary guesses a number from 1 to 4. a) What is the probability the numbers are equal? b) What is the probability the numbers are not equal?

Example 10  The sample space has 16 outcomes: { (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) Let E be the event the numbers are equal.  P(E) = 4/16 = ¼ b) Ē is the event the numbers are unequal.  P(Ē) = 1 – ¼ = ¾

Example 11  A diagram of a sample space S for an experiment is shown.

Example 11, cont’d  Find the probability of each of the events:  S   A  B  C 

Example 11, cont’d

Properties of Probability, cont’d 4) If events A and B are mutually exclusive, then 5) If A and B are any events, 6) For any event A and its complement:

Example 12  An experiment consists of spinning the spinner once and recording the number on which it lands.

Example 12, cont’d  4 events:  A: an even number  B: a number greater than 5  C: a number less than 3 a) Find P(A), P(B), and P(C). b) Find P(A U B) and P(A ∩ B). c) Find P(B U C) and P(B ∩ C).

Example 12, cont’d  The sample space has 8 outcomes: {1, 2, 3, 4, 5, 6, 7, 8}. a) Find P(A),P(B), and P(C).  A = {2, 4, 6, 8}, so P(A) = 4/8 = 1/2  B = {6, 7, 8}, so P(B) = 3/8  C = {1, 2}, so P(C) = 2/8 = 1/4

Example12, cont’d b) Find P(A U B) and P(A ∩ B).  A and B are not mutually exclusive, so  A ∩ B = {6, 8}, so P(A ∩ B) = 2/8 

Example 12, cont’d c) Find P(B U C) and P(B ∩ C).  B and C are mutually exclusive, so and 

Homework  Pg. 645  10, 16, 32, 38, 42