1. Aim: What is the multiplication rule?

2. Independent Event
Independent Event: the probability of an event occurring is not affected by the previous event
Example: flipping a coin, having a boy/girl

3. What is the multiplication rule?
Multiplication Rules can be used to find the probability of two or more events that occur in a sequence
Example: if a coin is tossed and then a die is rolled, one can find the probability of getting a head on he coin and a 4 on the die
These two events are independent since the outcome of the first event does not affect the probability outcome of the second event!

4. Multiplication Rule 1
When two events are independent, the probability of both occurring is
P(A and B) = P(A) * P(B)
Example: a coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die.
P(head and 4) = P(head) * P(4)

5. Extending to three events…
Multiplication Rule 1 can be extended to 3 or more events:
P(A and B and C and …) = P(A) * P(B) * P(C) * …

6. The “fifth” rule of probability!
There is a ‘fifth’ rule for independent pairs of events
Rule 5: Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent
P(A and B) = P(A)×P(B)
Sometimes called the multiplication rule for independent events.

7. Example
Since the 4 outcomes above are equally likely, then the probability of the second flip being heads is not affected by the result of the first flip.
In each case, the probability of heads is still ½ (doesn’t matter which column you are in, there are still two outcomes with one satisfying the condition of 2nd flip being heads.

8. Dependent Events
When the occurrence of the first event changes the probability of the occurrence of the second event
Example: suppose a card is drawn from a deck and it is not replaced and then a second card is drawn

9. Finding the probability of Dependent Events
To find the probability of dependent events, use the multiplication rule with a modification in notation…known as conditional probability
The notation for conditional probability is P(B|A)
Does not mean B is divided by A
It means that probability that event B occurs given that event A has already occurred

10. Simple Example
Find the probability that an ace is picked from a deck of cards and is not replaced when the second card is picked and it is king.
P(ace) * P(king without replacement)

11. Multiplication Rule 2
When two events are dependent of each other, the probability of both occurring is
P(A and B) = P(A) * P(B|A)

12. Class Work
An urn contains 3 red balls, 2 blue balls and 5 white balls. A ball is selected and its color noted. Then it is replaced. A second ball is selected and its color noted. Find the probability of each of these.
Selecting 2 blue balls
Selecting 1 blue and then 1 white ball
Selecting 1 red ball and then 1 blue ball
A Harris poll found 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 that they suffer great stress at least once a weak.
A person owns a collection of 30 CDs, of which 5 are country music. If 2 CDs are selected at random, find the probability that b both are country music.
Three cards are drawn from an ordinary deck and not repalced. Find the probability of these:
Getting 3 jacks
Getting an ace, a king, and a queen in order
Getting a club, a spade, and a heart in order
Getting 3 clubs
Box 1 contains 2 red balls and 1 blue ball. Box 2 contains 3 blue balls and 1 red ball. A coin is toosed. If it falls heads up, box 1 is selected and a ball is drawn. If it falls tails up box 2 is selected and a ball is drawn.
Draw a tree diagram showing the situation
What is the sample space
Find the probability of selecting a red ball.