Course 2, Unit 8 - Probability Distributions

Overview
Probability Distributions is the second unit on probability. In Patterns in Chance in Course 1, students learned to use the Addition Rule to find the probability that event A happens or event B happens. In the first lesson of this unit, students will learn to use the Multiplication Rule to find the probability that event A happens and event B happens. In Patterns in Chance, students used simulations to approximate probability distributions, such as the waiting-time distribution for the number of flips of a coin needed to get a head. In this unit, they will learn to use the Addition and Multiplication Rules to construct waiting-time distributions exactly.

After students complete this second probability unit in Core-Plus Mathematics, they will know the most important basic concepts of probability: sample spaces, equally likely outcomes, simulation and use of random digits, Addition Rule for Mutually Exclusive Events, the general Addition Rule, the Law of Large Numbers, the definition of conditional probability, the Multiplication Rule for independent events, the Multiplication Rule for dependent events, and expected value. Infinite series are introduced in the Lesson 3 Extensions tasks.

Key Ideas from Course 2, Unit 8

• Addition rule: If A and B are two events, P(A or B) = P(A) + P(B) - P(A and B). (This rule was developed in Course 1 Unit 8 on pp. 536-540.)

• Conditional probability: The probability that an event A occurs given that another event B occurs, written P(A | B). When P(B) > 0, P(A | B) = P(A and B)/P(B). (See pp. 528-529 and the example used in class for Problem 1. Knowing that a student is a girl likely changed the probability that the student was wearing sneakers.)

• Independent events: Two events A and B are independent if the occurrence of one of the events does not change the probability that the other event occurs. That is, P(A | B) = P(A). Alternatively, events A and B with nonzero probabilities are independent if P(A and B) = P(A) • P(B). For example, the probability of rolling doubles on each roll is 1/6. It does not matter whether or not previous rolls came up doubles. (See pp. 523-526.)

• Mutually exclusive events (or disjoint events): events that cannot occur on the same outcome. (See p. 530.)

• Multiplication rule for independent events: When A and B are independent events, the probability of A and B written as P(A and B) equals P(A) • P(B).

• Multiplication rule: If A and B are two events, P(A and B) = P(A)P(B | A). (See pp. 532-533.)

• Expected value: the mean of the distribution or Σx • P(x). (See pp. 545-548.)

• Waiting-time distributions: A waiting-time distribution occurs in situations in which someone is watching a sequence of independent trials and waiting for a certain event to occur. For example, the trials could be a person trying to shoot baskets and waiting for success. The shooter could be successful on the first try, or the shooter might have to wait for 10 shots for success to happen. The observer records the frequency with which the event occurred on the first trial, second trial, third trial, etc., in a frequency table. (See Course 1 Unit 2 for basic work with frequency tables.) (See pp. 560-564.)

  When First Makes Basket Number of Shots Frequency First try 1   Second try 2   Third try 3   Fourth try 4   ... ...
  Total	100

• Graph of a waiting-time distribution: has a characteristic shape. For example, suppose the probability of success is 40% on the first trial, then 40% of the time you will be successful on the first trial. It will take 2 trials only if the first observation was a failure and the second was a success. The probability of fail then success is (0.60)(0.40) = 0.24. (See pp. 560-564.) The graph will look as follows: