Conditional probability and the multiplication rule

1. Conditional Probability and the Multiplication Rule

Unit 3.3
CONDITIONAL PROBABILITY AND
THE MULTIPLICATION RULE

2. Multiplication Rule

The Multiplication Rule can be used to find
the probability of two or more events that
occur in a sequence .
The multiplication Rule for the probability of
A and B
If events A and B are independent, then the rule
can be simplified to P(A and B) = P (A) ● P (B). This
simplified rule can be extended for any number of
independent events.

3. Multiplication Rule Tip

1. Find the probability the first event occurs.
2. Find the probability the second event occurs
given the first event has occurred and
3. Multiply these two probabilities.

4. Using the Multiplication Rule to find Probability

1. A coin is tossed and a die is rolled. Find the
probability of getting a head and then rolling
a 6.

5. Using the Multiplication Rule to find Probability

1. A card is drawn from a deck and replaced;
then a second card is drawn. Find the
probability of selecting a Ace and then
selecting a queen.

6. Using the Multiplication Rule to find Probability

1. The probability that a salmon swims
successfully through a dam is 0.85. Find the
probability that two salmon successfully
swim through the dam.

7. Using the Multiplication Rule to find Probability

1. Two cards are selected from a standard deck
without replacement. Find the probability
that both are hearts.

8. Using the Multiplication Rule to find Probability

1. A Harris poll found the 46% of Americans
say they suffer great stress at least once a
week. If three people are selected at
random, find the probability that all three
will say they suffer great stress at least once
a week.

9. Using the Multiplication Rule to find Probability

1. The probability that a salmon swims
successfully through a dam is 0.85. Find the
probability that three salmon swim
successfully through the dam.

10. Using the Multiplication Rule to find Probability

1. Find the probability that none of the three
salmon are successful.

11. Using the Multiplication Rule to find Probability

1. Find the probability that at least one of the
three salmon is successful in swimming
through the dam.

12. Dependent Events

When the outcome or occurrence of the first
event affects the outcome or occurrence of
the second event in such a way that the
probability is changed, the events are said to
be dependent events.
Examples
Drawing a card from a deck, NOT replacing it, and
then drawing a second card.
Being a lifeguard and getting a tan.
Having high grades and getting a scholarship

13. Conditional Probability

To find probabilities when events are
dependent, use the multiplication rule with a
The probability of B
modification in notation.
given that event A
P(A and B) = P (A) ● P (B\A). has already occured

14. Finding Conditional Probability

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 that the king is not
replaced)
Solution: Because the first card is a king and
is not replaced, the remaining deck has 51
cards, 4 of which are queens. So,
P(B|A) = 4/51 = 0.078

15. Finding Conditional Probability

Three cards are drawn from an ordinary deck
and not replaced. Find the probability of
these events.
Getting 3 Jacks
Getting an ace, a king, and a queen in order
Getting a club, a spade, and a heart in order
Getting three clubs

16. Finding Conditional Probability

Gene
Present
Gene
Not
Present
Total
High IQ
33
19
52
Normal
IQ
39
11
50
Total
72
30
102
The table at the left 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.
Solution: There are 72
children who have the gene.
So, the sample space consists
of these 72 children, as shown
at the left. Of theses, 33 have a
high IQ. So,
P(B\A)= 33/72 =0.458

17. Finding Conditional Probability

Gene
Present
Gene
Not
Present
Total
High IQ
33
19
52
Normal
IQ
39
11
50
Total
72
30
102
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.