Lesson
9 Problem Solving:
When One Thing Depends on Another Thing
Problem Solving: When One Thing Depends on Another Thing
What happens when you don’t put the card
back in the deck?
We worked with coins, dice, and cards. We asked what our chances were
for drawing an ace out of a deck of cards that was shuffled. Remember,
the cards have to be mixed up, or random, for each card to have an equal
chance of being drawn from the deck. We know there are four aces in a
deck of 52 cards, which means that the probability is 4 out of 52.
Outcomes with drawing an ace: 4
Total possibilities: 52
4
The probabilitiy is 52
= 0.077, or about 8%.
Now let’s think about the chances of pulling two aces in a row out of a
deck and putting them face up on a table. We need to think about this
problem carefully because we are not going to put the card from the
first draw back in the deck.
Example 1 shows how we solve this problem. We already know the
chances of drawing an ace out of the deck on the first draw are
4
52 , or 0.077.
Unit 7 • Lesson 9 523
Lesson 9
Example 1 shows that we would only have 51 cards left in the deck
when we try to draw the second ace. In this case, drawing the second
ace depends on drawing the first ace. That means that the probability
of drawing the second ace depends on the card already drawn from the
deck. The probability changes slightly because we have one less card for
the second draw. We multiply the two fractions together to show what
the probability of drawing two aces in a row would be.
Example 1
Show the probability of drawing two aces in a row if the card from
the first draw is not put back in the deck.
Chances of selecting an ace on the first draw:
• Number of aces: 4
• Total possibilities: 52
Chances of selecting another ace on the second draw:
• Number of aces: 3
• Total possibilities: 51
Probability of drawing two aces in a row:
4
3
12
52 · 51 = 2,652 , or 0.0045
The answer 0.0045 is about 0.5%, which is less than 1%. This is a
small number.
Pulling colored marbles from a jar is another way to think about this
kind of probability.
524 Unit 7 • Lesson 9
Lesson 9
Suppose we had a jar with 10 marbles—6 green and 4 yellow. What are
the probabilities of first pulling out a green marble and then pulling out
a yellow marble?
The order is important. We need to pull out a green marble first, and
then a yellow one. Pulling out the yellow marble depends upon what we
do first.
Example 2 shows how we calculate the probabilities. Again, the total
number of marbles changes based on the first draw. That means on
the second draw there will still be 4 yellow marbles, but only 9 marbles
altogether.
Example 2
Show the probability of pulling out a green marble and then a
yellow marble.
Chances of drawing a green marble on the first draw:
• Number of green marbles: 6
• Total possibilities: 10
Chances of drawing a yellow marble on the second draw:
• Number of yellow marbles: 4
• Total possibilities: 9
Probability of drawing a green marble and then a yellow marble in
that order:
6 4 24
10 · 9 = 90 , or 0.266
Probability of drawing
a green marble.
Probability of drawing
a yellow marble.
The decimal number 0.266 is about 27%.
The probability is about 27%.
Unit 7 • Lesson 9 525
Lesson 9
What happens if we use just a few cards from
the deck?
We have been able to establish probabilities based on a full deck of
cards. A regular deck of cards has 52 total cards and a unique mix of
characteristics, numbers, colors, suits, etc. What happens if we use just
a few cards from the deck? We would have to come up with a new set of
probabilities based on the characteristics of the new set of cards.
Suppose we have a deck of just 5 cards. The cards are the 2 of hearts, 2
of diamonds, 3 of clubs, 4 of spades, and 5 of clubs.
�����
• What is the probability of drawing two red cards if we replace the
card drawn each time?
• What is the probability if we do not replace the first card drawn?
To answer these questions, we have to think about probabilities in a
different way.
We have two red cards and three black cards, for a total of five cards.
The first time we draw, we have a 2
5 chance of getting a red card. If we
2 2
4
replace that card, we have a 5 · 5, or 25
, or 16% chance of selecting
two red cards in a row.
1
2
If we do not replace the first card, we have a 2
5 · 4, or 20, or 10%
chance of selecting two red cards.
This is different from the probabilities we computed for a regular deck
of cards. The probabilities for that deck are 50% for selecting two red
cards if we replace the first card drawn, and 24.5% for selecting two
red cards if we do not replace the first card drawn.
Problem-Solving Activity
Turn to Interactive Text,
page 273.
526 Unit 7 • Lesson 9
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
We have created
a new deck of
cards with new
probabilities.
Lesson 9
Homework
Activity 1
Rewrite the numbers using scientific notation. Remember to round.
1. 0.222 ​
2. 5,876 ​
3. 0.0000000123 ​
4. 0.00045 ​
5. 17,500 ​
6. 191,000 ​
Activity 2
Each of the numbers has an error. They do not follow the rules for scientific
notation. Write the letter on your paper (a and/or b) that tells the reason why
it is not correct.
Model11.5 × 102
(a) The decimal number is not between 1 and 10
(b) The base of the power is not 10
Answer: a. The number is not between 1 and 10.
1. 2.7 × 43 ​
(a) The decimal number is not between 1 and 10.
(b) The base of the power is not 10.
2. 17.9 × 103 ​
(a) The decimal number is not between 1 and 10.
(b) The base of the power is not 10.
3. 44.7 × 10−4 ​
(a) The decimal number is not between 1 and 10.
(b) The base of the power is not 10.
4. 3.8 × 5−3 ​
(a) The decimal number is not between 1 and 10.
(b) The base of the power is not 10.
Copyright 2010 by Cambium Learning Sopris West®. All rights reserved. Permission is granted to reproduce this page for student use.
Unit 7 • Lesson 9 527
Lesson 9
Homework
Activity 3
A deck of cards is made up of the following:
3 kings—king of hearts, king of diamonds, king of spades
4 queens—(all four suits)
2 jacks—jack of hearts and jack of diamonds
4 aces—(all four suits)
Tell the probability of each of the following.
1. What is the chance of selecting 2 queens from the deck if you replace the
first card before drawing the second?
2. What is the chance of drawing a red card from the deck?
3. What is the chance of drawing a king of hearts?
4. What is the chance of drawing a king of hearts or a jack of diamonds from
the deck?
5. What is the chance of drawing 2 jacks from the deck if you do not replace the
first card before drawing the second?
6. What is the chance of drawing 2 red cards from the deck if you do not
replace the first card before drawing the second?
4
169
6
13
1
13
2
13
1
78
5
26
Activity 4 • Distributed Practice
Solve.
1.
15
5
÷ 13 2. 9.9 · 0.9 3. 198.32 + 227.05 + 164.99 + 201.87 4.
5.
5
12
12
1
1
8 − 4 4
3
4
+ 23 6. Convert 0.34 to a fraction and a percent. 4
17
50
7. Convert 5 to a decimal number and a percent. 8. Convert 100% to a fraction and a decimal number. 528 Unit 7 • Lesson 9
1
1
Copyright 2010 by Cambium Learning Sopris West®. All rights reserved. Permission is granted to reproduce this page for student use.