CHAPTER 14 WORKSHEET

We are rolling two four-sided dice having the numbers 1, 2, 3, and 4 on their faces. Outcomes in the sample space are pairs such as (1,3) and (4,4)

A) How many elements are in the sample space?

B) What is the probability that the total showing is even?

C) What is the probability that the total showing is greater than six?

SOLUTIONS

A) 16B) .5C) 3/16

An experimenter testing for extrasensory perception has five cards with pictures of a (s)tar, a (c)ircle, (w)iggly lines, a (d)ollar sign, and a (h)eart. She selects two cards without replacement. Outcomes in the sample space are represented by pairs such as (s,d) and (h,c).

A) How many elements are in this sample space?

B) What is the probability that a star appears on one of the cards?

C) What is the probability that a heart does not appear?

SOLUTIONS

A) 20B) 2/5C) 3/5

For the next problems;

a) Find the probability of the given event.

b) Find the odds against the given event.

FORMULA

𝑃ሺ𝐸ሻ= 𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝐸 𝑜𝑐𝑐𝑢𝑟𝑠𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 𝑖𝑠 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑒𝑑

PROBABILITY FORMULA FOR COMPUTING ODDS

If E’ is the complement of the event E, then the odds against E are

)(

)'(

EP

EP

QUESTIONS

A total of three shows when we roll two fair dice.

SOLUTIONS

a)

b) First find P(E’)

Then find

18

1

36

2)( EP

18

17

18

11)'( EP

1

17

181

1817

)(

)'(

EP

EP

We draw a face card when we select 1 card randomly from a standard 52-card deck.

2) a)

b) 10 to 3

13

3

52

12)( EP

3

10

133

1310

133133

1

)(

)'(

EP

EP

ASSUME THAT WE ARE DRAWING A 5-CARD HAND FROM A STANDARD 52-CARD DECK.

What is the probability that all cards are face cards?

We have to remember the counting technique C(52,5) ways to select a 5-card hand from a 52-card deck.

COMBINATION

Def. If we choose r objects from a

set of n objects, we say that we are forming a combination of n objects taken r at a time.

Notation C(n,r) = P(n,r) / r! = n! / [r!(n-r)!]

00030473.0960,598,2

792

)5,52(

)5,12(

C

C

What is the probability that all cards are red?

0.025

In a given year, 2,048,861 males and 1,951,379 females were born in the United States. If a child is selected randomly from this group, what is the probability that it is a female.

SOLUTION

Do you remember how to solve this problem?

0.04878

Males Females

Females

You are playing a game in which a single die is rolled. Calculate the expected value for each game. Is the game fair? See next slide for question.

If an odd number shows up, you win the number of dollars showing on the die. If an even number comes up, you lose the number of dollars showing on the die.

6,6

1

,5,6

1

,4 ,6

1

,3,6

1

,2 ,6

1

,1 ,6

1

66

55

44

33

22

11

VP

VP

VP

VP

VP

VP

The game is not fair.

2

1

6

12

6

9

6

6

6

5

6

4

6

3

6

2

6

1

66

1...3

6

12

6

1 1

6

1

You are playing a game in which a single die is rolled. If a four or five comes up, you win $2; otherwise, you lose $1.

0, the game is fair.

For the following problem, first calculate the expected value of the lottery. Determine whether the lottery is a fair game. If the game is not fair, determine a price for playing the game that would make it fair.

Five hundred chances are sold at $5 apiece for a raffle. There is a grand prize of $500, two second prizes of $250, and five third prize of $100.

5 ,500

492

95 ,500

5

245 ,500

2

495 ,500

1

44

33

22

11

VP

VP

VP

VP

NOW CALCULATE THE EXPECTED VALUE.

$3 to make the game fair.

292.495.98.99.