Download Probability of Simple Events

Probability of Simple Events
When tossing a coin, there are two possible outcomes, heads and tails. Suppose you are looking for heads.
If the coin lands on heads, this would be a favorable outcome. The chance that some event will happen (in
this case, getting heads) is called probability. You can use a ratio to find probability. The probability of an
event is a number from 0 to 1, including 0 and 1. The closer a probability is to 1, the more likely it is to
happen.
Example 1
There are four equally likely outcomes on the spinner.
Find the probability of spinning green or blue.
P(green or blue) =
number of favorable outcomes
number of total outcomes
2
4
= or
1
2
1
The probability of landing on green or blue is 2 , 0.50, or 50%.
Complementary events are two events in which either one or the other must happen, but both cannot
happen at the same time. The sum of the probabilities of complementary events is 1.
Example 2
There is a 25% chance that Sam will win a prize. What is the probability that Sam will not win a prize?
P(win) + P(not win) = 1
0.25 + P(not win) = 1
–0.25
= –0.25
P(not win) = 0.75
3
So, the probability that Sam won’t win a prize is 0.75, 75%, or 4.
Exercises
1. There is a 90% chance that it will rain. What is the probability that it will not rain?
One pen is chosen without looking from a bag that has 3 blue pens, 6 red, and 3 green. Find the
probability of each event. Write each answer as a fraction, a decimal, and a percent.
2. P(green)
3. P(blue or red)
4. P(not red)
Skills Practice- Probability of Simple Events
A card is randomly chosen. Find each probability. Write
each answer as a fraction, a decimal, and a percent.
1. P(B)
2. P(Q or R)
3. P(vowel)
4. P(consonant or vowel)
5. P(consonant or A)
The spinner shown is spun once. Write a sentence
explaining how likely it is for each event to occur.
7. P(dog)
9. P(dog or cat)
11. P(mammal)
The weather reporter says that there is a 12% chance that it will be moderately windy tomorrow.
12. What is the probability that it will not be windy?
13. Will tomorrow be a good day to fly a kite? Explain.
Example 2
In a telephone poll, 225 people were asked for
whom they planned to vote in the race for mayor.
What is the experimental probability of Juarez
getting a vote from a person selected at random?
Of the 225 people polled, 75 planned to vote for Juarez.
So, the experimental probability is
75
225
or
Juarez
Number of
People
75
Davis
67
Abramson
83
Candidates
1
.
3
Example 3
Suppose 5,700 people vote in the election. How many can be expected to vote for Juarez?
1
3
· 5,700 = 1,900
About 1,900 will vote for Juarez.
Exercises
1. PETS Use the graph of a survey of 150 students asked whether they prefer cats or dogs.
a. What is the experimental probability of a student preferring dogs?
b. Suppose 100 students were surveyed. How many can be
expected to prefer dogs?
c. Suppose 300 students were surveyed. How many can be expected to prefer cats?
4.
Practice
Fundamental Counting Principle
Use the Fundamental Counting Principle to find the total number of outcomes in each situation.
1. choosing from 8 car models, 5 exterior paint colors, and 2 interior Colors
2. selecting a year in the last decade and a month of the year
3. picking from 3 theme parks and 1-day, 2-day, 3-day, and 5-day passes
4. choosing a meat and cheese sandwich from the list shown in the table
5. tossing a coin and rolling 3 number cubes
Cheese
Meat
Provolone
Salami
Swiss
Turkey
American
Tuna
Cheddar
Ham
6. selecting coffee in regular or decaf, with or without cream, and with or without sweeteners
7 COINS Find the number of possible outcomes if 2 quarters, 4 dimes, and 1 nickel are tossed.
Independent and Dependent Events
The probability of two independent events can be found by multiplying the probability of the first event by the
probability of the second event.
Example 1
Two number cubes, one red and one blue, are rolled. What is the probability that the outcome of the
red number cube is even and the outcome of the blue number cube is a 5?
1
2
P(red number cube is even) =
P(blue number cube is a 5) =
1
6
1
1
1
P(red number cube is even and blue number cube is a 5) = 2  6 or 12
1
The probability that the two events will occur is 12.
If two events, A and B, are dependent, then the probability of both events occurring is the product of the
probability of A and the probability of B after A occurs.
Example 2
There are 6 black socks and 4 white socks in a drawer. If one sock is taken out without looking and
then a second is taken out, what is the probability that they both will be black?
P(first sock is black) =
6
10
or
5
P(second sock is black) = 9
3
5
6 is the number of black socks; 10 is the total number of socks.
5 is the number of black socks after one black sock is removed; 9
is the total number of socks after one black sock is
removed.
3
5
1
P(two black socks) = 5  9 or 3
1
The probability of choosing two black socks is 3 .
Exercises
A card is drawn from a deck of 10 cards numbered 1 through 10 and a number cube is rolled.
Find each probability.
1. P(10 and 3)
2. P(two even numbers)
5. P(two numbers less than 4)
There are 4 red, 6 green, and 5 yellow pencils in a jar. Once a pencil is selected, it is not replaced.
Find each probability.
7. P(red and then yellow)
8. P(two green)
9. P(green and then yellow)
Homework Practice
Independent and Dependent Events
The two spinners at the right are spun. Find each probability.
1. P(4 and C)
4. P(odd and A)
5. P(greater than 3 and B)
GAMES There are 10 yellow, 6 green, 9 orange, and 5 red cards in a stack of cards turned facedown.
Once a card is selected, it is not replaced. Find each probability.
7. P(two yellow cards)
8. P(a red card and then an orange card)
11. P(two cards that are not orange)
12. P(two cards that are neither red nor green)
14. BASKETBALL Angelina makes 70% of her free throws. What is the probability that she will make her
next two free throws?
15. CAR RENTALS Use the following information and the information in the table.
At a car rental office, 63% of the customers are men and 37% are
women.
a. What is the probability that the
next customer will be a woman
who requests a convertible?
b. What is the probability that the next
customer will be a man who
requests either a compact car or
luxury car?
Car Requests
Compact
25%
Full-size
37%
Convertible
10%
SUV
16%
Luxury
12%
Permutations
A permutation is an arrangement, or listing, of objects in which order is important. You can use the
Fundamental Counting Principle to find the number of possible arrangements.
Example 1
BOOKS How many ways can 4 different books be arranged on a bookshelf?
This is a permutation. Suppose the books are placed on the shelf from left to right.
There are 4 choices for the first book.
There are 3 choices that remain for the second book.
There are 2 choices that remain for the third book.
There is 1 choice that remains for the fourth book.
4•3•2•1
= 24
Simplify.
So, there are 24 ways to arrange 4 different books on a bookshelf.
Example 1
Find P(5,4).
4 • 3 • 2 • 120
Simplify.
Exercises
Find each value. Use a calculator if needed.
1. P(3,2)
2. P(7,6)
5. How many ways can you arrange the letters in the word group?
3. P(6,3)
Practice - Permutations
Solve each problem.
1. NUMBERS How many different 2-digit numbers can be formed from the digits 4, 6, and 8?
Assume no number can be used more than once.
2. LETTERS How many permutations are possible of the letters in the word numbers?
3. PASSENGERS There are 5 passengers in a car. In how many ways can the passengers sit in the 5
passenger seats of the car?
4. . TIME Michel, Jonathan, and two of their friends each ride their bikes to school. If they have an
equally-likely chance of arriving first, what is the probability that Jonathan will arrive first and Michel
will arrive second?
CODES For Exercises 5-7, use the following information. A bank gives each new customer a 4-digit code
number which allows the new customer to create their own password. The code number is assigned
randomly from the digits 1, 3, 5, and 7, and no digit is repeated.
5. What is the probability that the code number for a new customer will begin with a 7?
6. What is the probability that the code number will not contain a 5?
7. What is the probability that the code number will start with 371?