Algebra 2
Probability Notes #3: The Multiplication Rule
Name ______________________________
MAFS.912.S-CP.1.1
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the
outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
MAS.912.S-CP.1.5
Recognize and explain the concepts of conditional probability and independence in everyday language and everyday
situations.
The Multiplication Rule is for finding probabilities such as P(A and B), the probability that event A occurs
AND event B occurs.
 The key word to note here is “and”, meaning both occur.
Example: Randomly choose 2 shapes, but replace the first one before choosing the second:
1. Find P(triangle, then square) =
2. Find P(circle, then triangle) =
3. Find P(black triangle, then white circle) =
4. Find P(square, then square) =
These examples are all ___________________________, which means that the occurrence or nonoccurrence
of one event _________ ______ change the probability that the other event will occur.
Now let’s look at some ________________ events: when the occurrence or nonoccurrence of one event
___________ change the probability that the other event will occur.
Example: Randomly choose 2 shapes, but do NOT replace the first one before choosing the second:
1. Find P(triangle, then square) =
2. Find P(circle, then triangle) =
3. Find P(black triangle, then white circle) =
4. Find P( square, then square) =
Sometimes you can decide whether or not events are independent by using common sense.
Example 1:
Andrew lives in Texas, and the probability that he
will be alive in 10 years is 0.72. Ellen lives in New
York and the probability that she will be alive in 10
years is 0.92.
c) Do we have any reason to believe that the life
span of one of these people will affect the life
span of the other? Does this make them
dependent or independent?
d) Find the probability that both Andrew and Ellen
will be alive in 10 years.
Example 2:
A bag contains 18 suckers. 7 are cherry, 5 are
orange, and 6 are grape. Sheldon and Aaron are
drawing suckers randomly from the bag. Sheldon
randomly picks a grape sucker and eats it, then
Aaron randomly picks a cherry one.
a) Will Sheldon’s choice have an affect on the
probability of Aaron getting a cherry sucker?
Explain.
b) Find the probability of the event.
Other helpful hints when working with the multiplication rule:
 It is common practice to treat events as _______________ when small samples are drawn from
large populations. The guideline is:
If a sample size  ______%of the population, treat them as ________________
(even if selections are made without replacement)

To find the probability of “at least one”, find the ________________ of the probability of getting
one. Write the rule for this:
1. Which of these events are independent? Circle all that apply.
a) You flip a coin and get tails. You flip it a second time and get heads.
b) You pull your friend's name out of a hat that holds 20 different names, replace the name, then
draw out your friend's name again.
c) You spin a spinner divided into five equal parts numbered 1-5. You get a 3 on the first spin,
and then spin again and get a 2 on the second spin.
d) You remove a black sock from a drawer without looking, then remove another black sock.
2. You roll two fair dice, one green and one red.
a) Are the outcomes on the dice independent? Explain.
b) Find P(5 on the green die and 3 on the red die)
c) Find P(1 on the green die and 4 on the red die)
d) Find P(5 on the green die and 3 on the red die) or P(1 on the green die and 4 on the red die)
3. You select two cards at random from a standard deck of 52 playing cards without replacing the
first one before drawing the second. Find the following probabilities as simplified fractions.
a) P(2 hearts)
d) P(a face card, then a 2)
b) P(a club, then a spade)
e) P(a face card, then a face card)
c) P(a 3, then a 10)
f) Are these outcomes independent? Explain.
4. You select two cards at random from a standard deck of 52 playing cards, but before you draw the
second card, you put the first one back and reshuffle the deck. Find the following probabilities as
simplified fractions.
a) P(2 clubs)
d) P(a face card, then a 9)
b) P(a club, then a spade)
e) P(a black card, then a heart)
c) P(a 7, then a Queen)
f) Are the outcomes independent? Explain.
The Masterfoods company manufactures bags of Peanut Butter M&M's. They report that they make
10% each brown and red candies, and 20% each yellow, blue, and orange candies. The rest of the
candies are green. (Assume you have a large bag, so we assume independence).
5. If you pick two Peanut Butter M&Ms at random, what is the probability that:
a) Both are green?
c) Neither of them are orange?
b) You get a red, then a brown?
d) At least one of them is blue?
6. If you pick four Peanut Butter M&M's in a row randomly, what is the probability that
a) they are all blue?
c) at least one is red?
b) none are green?
d) the fourth one is the first one that is brown?
7. After picking 10 Peanut Butter M&M's in a row, you still have not picked a red one. A friend
says that you should have a better chance of getting a red one on your next pick since you have
yet to see one. Comment on your friend's statement.
8. Five multiple choice questions, each with four possible answers, appear on your history exam.
What is the probability that if you just guess, you
a) get none of the questions correct?
b) get all of the questions correct?
c) get at least one of the questions wrong?
d) get your first incorrect answer on the fourth question?
9. An automotive manufacturer buys computer chips from a supplier. The supplier sends a shipment
containing 5% defective chips (each chip chosen from this shipment has a probability of 0.05 of
being defective). If each automobile uses 12 chips selected independently, what is the probability
that all 12 chips in a car will work properly?
10. A teacher asks each student in the class to randomly choose a number from 1 to 50 and write it
down. Jerome and Denise each write down a number. Find the probability that both students’
choices will be greater than 35. Express your answer as a decimal rounded to the nearest
hundredth.