Statistics
3.2: Conditional Probability and the Multiplication Rule
Objective 1: I can find conditional probabilities.
In this section, you will learn to find the probability that 2 events occur in sequence. But before you
can do this, you must know how to find ________________________________. A conditional
probability is the probability of an event occurring given that _____________________________
_____________________. The conditional probability of event B given that event A has already
occurred is denoted _________ and is read “_________________________________________”.
Example 1:
A) Two cards are selected in sequence from a standard deck. Find the probability that the second
card is a queen, given that the first card is a king. (Assume the king is not replaced.)
Solution: Because the first card drawn was a king, there are still ____ queens in the deck. But the
king is not replaced which means there are only _____ cards remaining to choose from. This event
has the probability of
We would write that as P
B) The table below shows the results of a study in which researchers examined a child’s IQ and the
presence of a specific gene in the child. Find the probability that a child has a high IQ given that
the child has the gene.
gene
gene not
Total
present present
High IQ
33
19
52
Normal IQ
39
11
50
Total
72
30
102
Solution: There are _____ children who have the gene present (1st column). Out of those 72, ____
have a high IQ. So the probability of this event is
We would write that as
TIY 1:
1) Find the probability that a child does not have the gene.
2) Find the probability that a child does not have the gene, given that the child has a normal IQ.
3) Find the probability that a child has the gene present, given that they have a normal IQ.
Objective 2: I can distinguish between independent and dependent events.
In some experiments, one event does not affect the probability of another event. For example, when
you roll a die and flip a coin, neither event affects the outcome of the other. These events are called
_____________________________.
Two events are independent if _______________________________________________________
_____________________________.
Using probability notation: Two events, A and B, are independent if
*Events that are not independent are ________________________.
*To determine if A and B are independent, first calculate ____. Then calculate _________. If the
values are _______, then the events are ____________________. If
Example 2: Classify the events as independent or dependent events.
A) Selecting a king from a standard deck, not replacing it, and then selecting a queen from the deck.
Solution: P(B|A) =
and P(B) =
. Keeping the king out of the deck changes the probability for
drawing a queen. These events are _________________.
B) Tossing a coin and getting a head, and then rolling a 6-sided die and obtaining a 6.
Solution: P(B|A) =
and P(B) =
. Tossing the coin does not affect the outcome when you roll
the die. These events are __________________.
C) Driving over 85 miles per hour (A), and then getting in a car accident (B).
TIY 2: Decide whether the events are independent or dependent.
1) Smoking a pack of cigarettes per day and developing emphysema, a chronic lung disease.
2) Exercising frequently and having a 4.0 grade point average.
3) Rolling an even number on a die and then rolling a 4.
Objective 3: I can use the multiplication rule to find probabilities.
To find the probability of two events occurring in sequence, use the multiplication rule. The
probability that two events A and B will occur in sequence is
If events A and B are independent, then the rule can be simplified to
This rule can be extended for more than 2 events as well.
Example 3: Use the Multiplication Rule to find each probability.
1) Two cards are selected without replacement from a standard deck of cards. Find the probability
of selecting a King then selecting a Queen.
2) A coin is tossed and a die is rolled. Find the probability of getting a head, then rolling a 6.
TIY 3:
1) The probability that a salmon swims successfully through a dam is 0.85. Find the probability that
two salmon swim successfully through the dam.
2) Two cards are selected from a standard deck without replacement. Find the probability that both
cards are hearts.
3) Two cards are selected from a standard deck with replacement. Find the probability that both
cards are hearts.
4) Find the probability of rolling 2 even numbers in a row on a six-sided die.
Read Examples 4 and 5, pg 152 and 153.
TIY 4: The probability that the knee surgery is successful is now 90%.
1. Find the probability that three knee surgeries are successful.
2. Find the probability that at least one of the knee surgeries is successful.
TIY 5: In a jury pool, 65% of the people are female. Of these 65%, one out of four works in a
health field.
1. Find the probability that a randomly selected person from the jury pool is female and works in a
health field.
2. Find the probability that a randomly selected person from the jury pool is female and does not
work in a health field.
Extra Examples:
This is #13 from page 154.
In the general population, one woman in eight will
develop breast cancer. Research has shown that 1
woman in 600 carries a mutation of the BRCA gene.
Eight out of 10 women with this mutation develop
breast cancer.
This is #15 from page 155.
The table shows the results of a survey in which 146
families were asked if they own a computer and if they
will be taking a summer vacation this year.
A) Find the probability that a randomly selected family
is not taking a summer vacation this year.
A) Find the probability that a randomly selected
woman will develop breast cancer given that she has a
mutation of the BRCA gene.
B) Find the probability that a randomly selected
woman will carry the mutation of the BRCA gene and
will develop breast cancer.
C) Are the events of carrying this mutation and
developing breast cancer independent or dependent?
Explain.
B) Find the probability that a randomly selected family
owns a computer.
C) Find the probability that a randomly selected family
is taking a summer vacation this year given that they
own a computer.
D) Find the probability a randomly selected family is
taking a summer vacation this year and owns a
computer.
E) Are the events of owning a computer and taking a
summer vacation this year independent or dependent?
Explain.