Fundamentals of Probability Unit 7A Fundamentals of Probability Ms. Young

Definitions Outcomes are the most basic possible results of observations or experiments. An event consists of one or more outcomes that share a property of interest. Ms. Young

Expressing Probability The probability of an event, expressed as P(event), is always between 0 and 1 (inclusive). A probability of 0 means the event is impossible and a probability of 1 means the event is certain. 1 0.5 Certain Likely Unlikely 50-50 Chance Impossible 0 ≤ P(A) ≤ 1 Meteorology offers a rich source of ideas here. Would you take an umbrella to work if the chance for rain is at 10%? How about 90%? Ms. Young

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). Ms. Young

Outcomes and Events Assuming equal chance of having a boy or girl at birth, what is the probability of having two girls and two boys in a family of four children? Of the 16 possible outcomes, 6 have the event two girls and two boys. P(2 girls) = 6/16 = 0.357 One of the challenges many students have with probability is knowing when to trust intuition and when to back away from it. Many students would answer the question with a 50%. This problem is a classic example of the importance of looking at the total number of outcomes. Ms. Young

Empirical and Subjective Probabilities An empirical probability is based on observations or experiments. It is the relative frequency of the event of interest. A subjective probability is an estimate based on experience or intuition. Ms. Young

Three Types of Probabilities Theoretical probability The chance of rolling a 4 is 1 out of 6. Empirical probability Subjective probability Ms. Young

Three Types of Probabilities Theoretical probability She’s a 92% free throw shooter for the season. Empirical probability Subjective probability Ms. Young

Three Types of Probabilities Theoretical probability There’s about a 70% chance she will go out on a date with me. Empirical probability Subjective probability Ms. Young

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 1 – P(A). Since the probability of a family of four children having two girls and two boys is 0.375, what is the probability of a family of four children not having two girls and two boys? P(not 2 girls) = 1 – 0.375 = 0.625 Ms. Young

Making a Probability Distribution A probability distribution represents the probabilities of all possible events. To make a probability distribution, do the following: Step 1: List all possible outcomes. Use a table or figure if it is helpful. Step 2: Identify outcomes that represent the same event and determine the probability of each event. Step 3: Make a table listing each event and probability. Ms. Young

A Probability Distribution All possible outcomes and a probability distribution for the sum when two dice are rolled are shown below. Possible outcomes Point out that the reason 7 is considered a lucky number in many gambling games is simply because there are more ways to roll a 7 than any other number. Ms. Young

Odds Odds are the ratio of the probability that a particular event will occur to the probability that it will not occur. The odds for an event A are . The odds against an event A are . Ms. Young