No Warm-Up today. You have a Quiz Clear your desk of everything but a calculator and something to write with.

Unit 2 Day 5

Definitions Experiment – process that gives definite results Outcomes – results of an experiment Sample space – set of all possible outcomes

Examples Experiment – Tossing a coin – Outcomes: Heads or Tails – Sample space: S = {H, T} Experiment – Rolling a die – Outcomes: 1, 2, 3, 4, 5, and 6 – Sample space: S = {1, 2, 3, 4, 5, 6}

Example Consider this dartboard. Assume that the experiment is “throwing a dart” once and that the dart hits the board. Find each of the following. a) The outcomes b) The sample space Solution: a) The outcomes are hitting white (W), purple (P), or yellow (Y). b) The sample space is {hitting white, hitting purple, hitting yellow}, which can be stated as {W, P, Y}.

Definition of an Event If S is a sample space of an experiment, then an event is any subset of the sample space. Examples… 1.Die showing an even number 2.Picking an ace from a deck of cards

Example If an experiment consists of tossing a coin three times and recording the results in order, find the sample space. (Find all possible outcomes) {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

Example The event E of showing “exactly two heads” is the subset of S that consists of all outcomes with two heads. Write out that possible event. E = {HHT, HTH, THH}

Example What is the event F of showing “at least two heads”? F = {HHH, HHT, HTH, THH} What is the event G of showing “no heads”? G = {TTT}

Types of Probability Experimental Probability – Based on trials and observations Theoretical Probability – Calculated by analyzing a situation

Experimental Probability Sociological Survey. The authors of this text conducted an experimental survey to determine the number of people who are left- handed, right-handed, or both. The results are shown in the graph. a) Determine the probability that a person is right-handed. b) Determine the probability that a person is left-handed.

Solutions: a) The number of right-handed is 82, the number of left-handed is 17, the number of ambidextrous is 1. The total number of observations is = 100. The probability that a person is right-handed is P = 82/100 =.82 = 82% b) The probability that a person is left-handed is P, where P = 17/100 =.17 = 17%

Probability Facts P(E) is a number between 0 and 1, 0 ≤ P(E) ≤ 1 If an event is certain to occur, then P(E) = 1 If an event is impossible, then P(E) = 0 The closer the probability of event is to 1, the more likely the event is to happen.

Examples If you flip a coin, what is the theoretical probability that it lands with heads up? – ½ or 50% If you flip a coin, what is the theoretical probability that it lands with tails up? – ½ or 50% How would you find experimental probability?

Examples If you roll a standard die, what is the theoretical probability that it lands with the 3 facing up? – 1/6 If you roll a standard die, what is the theoretical probability that it lands with the 3 or the 4 facing up? – 2/6 or 1/3

Example Suppose we select, without looking, one marble from a bag containing 4 red and 9 purple marbles. What is the probability of selecting a red marble? Solution: There are 13 equally likely ways of selecting any marble, and 4 ways of selecting red. P(selecting a red marble) = 4/13

Example What is the probability of getting a sum of 5 on a roll of a pair of dice?

Rolling a Pair of Dice

Solution On each die, there are 6 possible outcomes. The outcomes are paired so there are 6(6) or 36 possible ways in which the two can fall. There are 4 ways to roll a total of 5: (1, 4) (4, 1) (2, 3) and (3, 2). P(sum of 5) = 4/36 = 1/9

Your Turn! What is the probability of choosing, at random, the ace of spades from a deck of 52 cards? What is the probability of choosing any ace from a deck of 52 cards? What is the probability of drawing a red card from a deck of 52 cards? What is the probability of drawing a club from a deck of 52 cards?

Solutions What is the probability of choosing, at random, the ace of spades from a deck of 52 cards? 1/52 What is the probability of choosing any ace from a deck of 52 cards? 4/52 = 1/13 What is the probability of drawing a red card from a deck of 52 cards? 26/52 = 1/2 What is the probability of drawing a club from a deck of 52 cards? 13/52 = 1/4

Example A five-card poker hand is drawn from a standard deck of 52 cards. What is the probability that all five cards are spades?

= picking 5 spades picking 5 cards = 13(12)(11)(10)(9) 52(51)(50)(49)(48) = or %

Solution How many different ways can we choose five spades from 13 spades? Probability of drawing five spades is

Example A bag contains 20 tennis balls, of which four are defective. If two balls are selected at random from the bag, what is the probability that both are defective? = pick a defected tennis ball picking any two tennis balls

Solution = 4(3) 20(19) = or %

Homework Probability WS