6.2 Basics of Probability LEARNING GOAL Know how to find probabilities using theoretical and relative frequency methods and understand how to construct basic probability distributions. Page 238

Definitions An event is a collection of one or more outcomes that share a property of interest. Outcomes are the most basic possible results of observations or experiments. Page 239

Expressing Probability The probability of an event, expressed as P(event), is always between 0 and 1 inclusive. A probability of 0 means that the event is impossible, and a probability of 1 means that the event is certain. Page 239 Figure 6.2 The scale shows various degrees of certainty as expressed by probabilities.

Theoretical Probabilities Theoretical Method for Equally Likely Outcomes Step 1. Count the total number of possible outcomes. Step 2. Among all the possible outcomes, count the number of ways the event of interest, A, can occur. Step 3. Determine the probability, P(A), from P(A) = Pages 239-240 number of ways A can occur total number of outcomes

EXAMPLE 1 Guessing Birthdays Suppose you select a person at random from a large group at a conference. What is the probability that the person selected has a birthday in July? Assume 365 days in a year. Page 240

Counting Outcomes Suppose process A has a possible outcomes and process B has b possible outcomes. Assuming the outcomes of the processes do not affect each other, the number of different outcomes for the two processes combined is a × b. This idea extends to any number of processes. For example, if a third process C has c possible outcomes, the number of possible outcomes for the three processes combined is a × b × c. Page 241

EXAMPLE 2 Some Counting How many outcomes are there if you roll a fair die and toss a fair coin? What is the probability of rolling two 1’s when two fair dice are rolled? Page 241

EXAMPLE 3 Counting Children What is the probability that, in a randomly selected family with three children, the oldest child is a boy, the second child is a girl, and the youngest child is a girl? Assume boys and girls are equally likely. Page 241

Theoretical Probabilities Relative Frequency Probabilities The second way to determine probabilities is to approximate the probability of an event A by making many observations and counting the number of times event A occurs. This approach is called the relative frequency (or empirical) method. Page 242 Slide 6.2- 9

Relative Frequency Method Step 1. Repeat or observe a process many times and count the number of times the event of interest, A, occurs. Step 2. Estimate P(A) by P(A) = number of times A occurred total number of observations Page 242 Slide 6.2- 10

EXAMPLE 4 500-Year Flood Geological records indicate that a river has crested above a particular high flood level four times in the past 2,000 years. What is the relative frequency probability that the river will crest above the high flood level next year? Page 242 Slide 6.2- 11

EXAMPLE 5 Which Method? Identify the method that resulted in the following statements. a. The chance that you’ll get married in the next year is zero. b. Based on government data, the chance of dying in an automobile accident is 1 in 7,000 (per year). c. The chance of rolling a 7 with a twelve-sided die is 1/12. Page 243 Slide 6.2- 12

Probability of an Event Not Occurring If the probability of an event A is P(A), then the probability that event A does not occur is P(not A). Because the event must either occur or not occur, we can write P(A) + P(not A) = 1 or P(not A) = 1 – P(A) Note: The event not A is called the complement of the event A; the “not” is often designated by a bar, so Ā means not A. Page 244 Slide 6.2- 13

Example: Rolling a die. What is the probability of getting 2 on the die? What is the probability of not getting 2 on the die?

Classwork 1. A standard deck of cards has 52 cards. a) What is the probability of drawing an ace from the shuffled deck of cards? b) What is the probability of drawing anything but an ace? 2. A bag contains 12 identically shaped blocks, 3 of which are red and the remainder are green. The bag is well shaken and a single block is drawn. a) What is the probability that the block is red? b) What is the probability that the block is not red?

EXAMPLE 8 Tossing Three Coins Make a probability distribution for the number of heads that occurs when three coins are tossed simultaneously. Tossing Three Coins Result Probability Page 246 Slide 6.2- 16