

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 waitingtime 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
waitingtime distributions exactly.
After students complete this second probability unit in CorePlus
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 pages 536540.)

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
pages 523526.)

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). (See
pages 523528.)

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 pages 528529 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.)

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

Multiplication rule: If A and B are two events, P(A and B) = P(A)P(B  A). (See
pages 532533.)

Expected value: the mean of the distribution or Σx • P(x). (See
pages 545548.)

Waitingtime distributions: A waitingtime 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 pages 560564.)
When
First Makes Basket

Number
of Shots

Frequency

First
try

1


Second
try

2


Third
try

3


Fourth
try

4


...

...






Graph of a waitingtime 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 pages 560564.) The
graph will look as follows:
