Download lesson 2-h - Oregon Focus on Math

COMPOUND PROBABILITIES USING
MULTIPLICATION & SIMULATION
LESSON 2-H
M
aya was making sugar cookies. She decorated them with one of two types of frosting (white or pink),
one of three types of sprinkles (chocolate, rainbow or green) and one of two types of candy (peppermint or
caramel).
The tree diagram and list show the possible outcomes for the types of cookies Maya made.
Chocolate
White
Rainbow
Green
Peppermint
Caramel
Possible Outcomes
Peppermint
White, Chocolate, Peppermint
White, Chocolate, Caramel
White, Rainbow, Peppermint
White, Rainbow, Caramel
White, Green, Peppermint
White, Green, Caramel
Pink, Chocolate, Peppermint
Pink, Chocolate, Caramel
Pink, Rainbow, Peppermint
Pink, Rainbow, Caramel
Pink, Green, Peppermint
Pink, Green, Caramel
Caramel
Peppermint
Caramel
Peppermint
Chocolate
Pink
Rainbow
Green
Caramel
Peppermint
Caramel
Peppermint
Caramel
Maya made a total of 12 different cookies using her ingredients. Notice that the total
number of outcomes is the product of the number of frosting options, sprinkle options
and candy options (2 ∙ 3 ∙ 2 = 12). The Multiplication Counting Principle relates the
number of choices to the number of outcomes. This principle helps identify the number
of outcomes without having to show the possible outcomes in a list, tree diagram or
table.
34 Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation
EXAMPLE 1
Ice cream sundaes at Gary’s Creamery come in five flavors with four possible
toppings. How many different sundaes can be made with one flavor of ice cream
and one topping?
Solution
Multiply the number of options for
ice cream with the number of options
for toppings.
4 ∙ 5 = 20
There are a total of 20 possible sundaes.
EXAMPLE 2
Oregon issues license plates consisting of three letters and three numbers. There
are 26 letters and the letters may be repeated. There are ten digits and the digits
may be repeated. How many possible license plates can be issued with three letters
followed by three numbers?
Solution
The license plate has six total letters and numbers. The first three are letters (A-Z)
followed by three numbers (0 – 9).
Multiply the possibilities.
26 ∙ 26 ∙ 26 ∙ 10 ∙ 10 ∙ 10 = 17,576,000
There are a total of 17,576,000 license plate options.
EXAMPLE 3
There are five students running a race. How many possible ways can they finish
first, second and third?
Solution
There are five students to choose from for first place. There will then only be four left
to choose from for second place and three left to choose from for third place.
Multiply the possibilities.
5 ∙ 4 ∙ 3 = 60
There are a total of 60 different ways the students could finish first, second and third.
When finding compound probabilities you must know the number of favorable outcomes and the number of
possible outcomes in the sample space. You can use a list, tree diagram, table or the Multiplication Counting
Principle to determine the number of favorable outcomes and possible outcomes.
Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation 35
EXAMPLE 4
Ross has a bag of marbles that has three red, four blue and five green marbles. He
chooses one marble, replaces it and then chooses a second marble. What is the
probability he chose a red marble and then a green marble?
Solution
Multiply to find the total number of
outcomes possible. There are 12 marbles
to choose from each draw.
12 ∙ 12 = 144
Multiply to find the number of favorable
outcomes. There are 3 possible red
marbles to choose from in the first draw
and 5 possible green marbles to choose
from in the second draw.
3 ∙ 5 = 15
Find the probability.
P(red then green) =
15
5
=
144 48
The probability Ross chose a red marble and then a green marble is
EXAMPLE 5
A multiple choice test has five questions. Each
question has four options to choose from. Marty
randomly guesses on every problem. What is the
probability he guessed correctly on each problem?
Solution
Multiply to find the total number of
outcomes possible. There are 4 choices
on each of the 5 questions.
4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 = 1024
Multiply to find the number of favorable
outcomes. There is 1 correct answer for
each of the 5 questions.
1∙1∙1∙1∙1=1
5
.
48
1
Find the probability.
P(guess correctly) = ____
​ 1024
   ​ ≈ 0.00098
The probability Marty guessed correctly on all the questions is about 0.00098 or
0.098%.
36 Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation
Sometimes it is difficult to make a list, tree diagram or table to show all the possible outcomes. Other times
the events depend on one another and so the Multiplication Counting Principle cannot give the number of
outcomes in the sample space or the number of favorable outcomes.
Simulations can give a good estimate for a probability when it is difficult to determine. A simulation is an
experiment you use to model a situation. You can use coins, number cubes, random number generators or
other objects to simulate events. The more trials you simulate, the better your estimate for a probability. The
Explore! shows a simulation.
EXPLORE!
PROBABILITY SIMULATION
Ainsley has a bag of marbles that has three red, four blue and five green marbles. She chooses one marble,
does not replace it and then chooses a second marble. What is the probability she chose a red marble and
then a green marble?
Step 1: To answer this question, you will perform a simulation. Place three red, four blue and five green
marbles in a bag or box. Copy the frequency table below onto your paper.
Red then Green?
Yes
No
Step 2: Without looking, choose a marble from the bag or box. Place it on your desk. Then choose a second
marble from the bag or box and set it on your desk. Make a tally in your frequency table under ‘Yes’
if you picked a red marble and then a green marble. Make a tally under ‘No’ if you did not pick a
red marble and then a green marble. Return the marbles to the bag.
Step 3: Continue the simulation until you have 10 results (tally marks). You have finished 10 trials.
Compute the experimental probability below. Write the probability as a percent.
frequency of red and then green
total number of trials
P(Red then Green) = ______________________
​    
  
 
​
Step 4: Continue the simulation 40 more times (total of 50 trials). Make a tally in the table for each result.
Compute the experimental probability again. Write the probability as a percent.
frequency of red and then green
total number of trials
P(Red then Green) = ______________________
​    
  
 
​
Step 5: Compare the experimental probabilities within the class. What is your estimate for the probability Ainsley chooses a red marble and then a green marble? Why?
Step 6: The theoretical probability that Ainsley will choose a red marble and then a green marble is
approximately 11%. How close is your estimated probability from the experiment? What might make your estimate be closer to the theoretical probability for choosing red then green?
Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation 37
EXERCISES
Find the number of possible outcomes for each situation.
1. A soccer team’s kit consists of two jerseys, two pairs of shorts and two pairs of socks. How many soccer
outfit combinations are possible if each outfit contains one jersey, one pair of shorts and one pair of socks?
2. Heather narrowed her clothing choices for the big party down to three skirts, two tops and four pairs of
shoes. How many different outfits are possible from these choices?
3. The ice cream shop offers 31 flavors. You order a double-scoop cone. If you want two different flavors,
how many different ways can the clerk put the ice cream on the cone?
4. The roller skating store sells girls’ roller skates with the following options:
Colors: white, beige, pink, yellow, blue
Sizes: 4, 5, 6, 7, 8
Extras: tassels, striped laces, bells
Assume all skates are sold with ONE extra. How many possible
arrangements exist?
5. A pizza shop offers 10-inch, 12-inch and 16-inch sizes with thin, thick,
deep dish or garlic crust. Also, the customer can choose one topping from
extra cheese, pepperoni, sausage, mushroom and green pepper. How
many pizza combinations are possible?
6. How many ways can six people stand in line at the movies?
7. One coin is tossed three times. How many outcomes are possible?
8. A phone number has seven numbers and starts with a 3-digit area code.
However, the 7-digit number cannot start with 0 (that calls the operator).
a. How many different 7-digit phone numbers are possible in each area code?
b. Why do some areas have more than one area code?
9. There are fifteen school bands participating in a competition. In how many ways
can first, second and third place be awarded?
Find each probability.
10. Four coins are tossed. What is the probability of tossing four heads?
11. In a school lottery, each person chooses a 3-digit number using any of the numbers 0 – 9 for each digit.
One 3-digit number is chosen from all possible 3-digit numbers. What is the probability of winning the
school lottery?
38 Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation
12. A bag contains 10 red marbles, 3 green marbles and 2 white marbles. Ed chooses one marble,
replaces it and then chooses another marble.
a. What is the probability he will choose two red marbles?
b. What is the probability he will choose a red marble and then a green marble?
c. What is the probability he will choose a green marble and then a white marble?
13. You roll a number cube three times. What is the probability of rolling a five every time?
14. Megan picks a card from a deck of cards numbered 1 – 10 and rolls a number cube. What is the
probability she chooses a card with a 5 and rolls an even number?
15. Natasha is given a four-digit password for her ATM account. Every
ATM password uses the digits 0 – 9 which can be repeated in the
password. What is the probability Natasha’s password is 1234?
16. Justin is given a password for his ATM account. It is a four-digit
password using the digits 0 – 9 and the digits cannot be repeated.
What is the probability Justin’s password is 1234?
17. An ice cream comes in either a cup or a cone and the flavors available are chocolate, strawberry and
vanilla. If you are given an ice cream at random, what is the probability it will be a cup of chocolate ice
cream?
18. What is the probability that you roll a number divisible by 3 on a number cube twice in a row?
19. You have cards with the letters C, S, M, I, U on them.
a. You pick one card, keep it and then pick the next card. This is repeated until all the cards are chosen.
What is the probability you pick the cards in the order M, U, S, I, C?
b. You pick one card, keep it and then pick the next card. This is repeated until all the cards are chosen.
What is the probability the first three cards are S, U, M, in that order?
c. You pick one card, replace it and then pick the next card. This is repeated until five cards are picked.
What is the probability you pick the cards in the order M, U, S, I, C?
d. You pick one card, replace it and then pick the next card. This is repeated until five cards are picked.
What is the probability the first three cards are S, U, M, in that order?
e. Do you have a better chance of picking the cards in the order M, U, S, I, C if you keep the cards or
replace them after each pick? Explain your answer.
20. Twenty-five percent of the jelly beans in a jar are orange. Yellow jelly
beans make up one-fifth of the total. Five percent are white. The other
half of the jar contains blue and green jelly beans.
a. What is the probability of picking an orange jelly bean, replacing it
and then picking a white jelly bean?
b. What is the probability of picking a yellow, then orange, then finally
a green or blue jelly bean if you replace the jelly beans after each
pick?
Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation 39
Perform each simulation to estimate the probability.
21. A new ice skating rink opened. The owner gave each person a red, blue or green glow bracelet. Suppose
there is an equal chance of getting any color of glow bracelet.
a. What is the theoretical probability of getting a blue glow bracelet?
b. Simulate this probability using a number cube. Let the numbers 1 and 2 represent a red glow
bracelet. Let the numbers 3 and 4 represent a blue glow bracelet. Let the numbers 5 and 6 represent
a green glow bracelet. Roll the number cube 50 times and place a tally under each “color” rolled in
each trial.
Red
Blue
Green
1 or 2
3 or 4
5 or 6
c. What is your experimental probability of getting a blue glow bracelet? How close is your
experimental probability to the theoretical probability in part a?
d. Suppose you actually have a 1 chance of getting a red glow bracelet, a 1 chance of getting a blue
3
6
glow bracelet and a 1 chance of getting a green glow bracelet. How would the simulation need to
2
change to reflect these probabilities? Explain your answer.
22. There is a 20% chance a person exposed to a virus will become sick. If you are exposed to the virus three
times, what is the probability you will become sick? Follow Steps 1-3 to simulate this situation.
Step 1: Use a random number generator on a calculator or place ten pieces of paper (numbered 1–10)
in a bag to randomly pull.
Step 2: Let the numbers 1 and 2 represent a person becoming sick.
Step 3: Show three random numbers from a calculator generator or pull three numbers from a hat,
replacing the number after each pick. If one of the numbers is a 1 or 2 you became sick. Put a
tally in the ‘Sick’ column. If all the numbers are 3–10 place a tally in the ‘Not Sick’ column.
Sick
40 Not Sick
a. Complete 50 trials (three pulls each) and record your tallies. What is the probability you will become
sick if you are exposed to the virus three times?
b. How would this simulation need to change if there was a 60% chance of getting sick after being
exposed to the virus?
c. How would this simulation need to change if there was a 75% chance of getting sick after being
exposed to the virus?
Lesson 2-H ~ Compound Probabilities Using Multiplication & Simulation