Download multiplication rule for independent events.

Aim: What is the multiplication
rule?
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
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!
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)
1 1 1
  
2 6 12
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) * …
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.
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.
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
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
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)
4 4
4
 
52 51 663
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)
Class Work
1.
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.
A.
B.
C.
2.
3.
4.
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:
A.
B.
C.
D.
5.
Selecting 2 blue balls
Selecting 1 blue and then 1 white ball
Selecting 1 red ball and then 1 blue ball
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.
A.
B.
C.
Draw a tree diagram showing the situation
What is the sample space
Find the probability of selecting a red ball.