GEOMETRIC PROBABILITY AND EXPECTED VALUE

JIM RAHN

WWW.JAMESRAHN.COM

A dart board consists of five concentric circles. The width of each of the rings is the same. If a

dart is randomly launched at the dart board and it lands on the board. What is the probability the

dart lands in each region?

A Warm Up Activity

A dart board consists of five concentric circles. The width of each of the rings is the same. If a dart is randomly thrown at the dart board and it lands on the board. What is the probability the dart lands in each region?

LABEL Area of Disk

A π(1)2 = 1π

B π(2)2 - π(1)2 = 3π

C π(3)2 - π(2)2 = 5π

D π(4)2 - π(3)2 = 7π

E π(5)2 - π(4)2 = 9π

Total π(5)2 = 25π

Theoretical Solution Using Geometry

Prob.

1/25=0.04

3/25=0.12

5/25=0.20

7/25=0.25

9/25=0.36

Can we simulate tossing a dart at this dart board through a probability experiment?

Place labeled disks (or cubes) with the indicated number of disks into a bag. Draw one disk at a time, record result, return disk to the bag. Repeat 100 times

ExperimentalResults

Label Label # of Disks in Bag

A 1

B 3

C 5

D 7

E 9

Total 25

Simulating the Tossing of the Darts

Tag A B C D E Total

#

Possible results

ExperimentalResults

Simulating the Tossing of the Darts

Tag A B C D E Total

# 4 15 15 29 37 100

Label Label # of Disks in Bag

A 1

B 3

C 5

D 7

E 9

Total 25

Simulating the Toss with the graphing calculator

Simulating the Tossing of the Darts- Method 2

Tag A B C D E Total

#

Select and classify 100 numbers

Label Label # of Disks in Bag

A 1

B 3

C 5

D 7

E 9

Total 25

ExperimentalResults

Use random number generator on calculator randInt(1,25,5)A = 1B = 2, 3, 4C = 5, 6, 7, 8, 9D = 10, 11, 12, 13, 14, 15, 16E = 17, 18, 19, 20, 21, 22, 23, 24, 25

Possible results

Simulating the Tossing of the Darts- Method 2

Tag A B C D E Total

# 4 13 23 28 32 100

Label Label # of Disks in Bag

A 1

B 3

C 5

D 7

E 9

Total 25

ExperimentalResults

A dart board consists of five concentric circles. The width of each of the rings is the same. If a dart is randomly launched at the dart board and it lands on the board. What is the probability the dart lands in each region?

FAIR GAME? Pay $1 to toss a dart. Hit A: Get back $3Hit B: Get back $2Hit C: Get back $1Hit D or E: Lose $1

Expected Value

Label Label # of Disks in Bag

A 1

B 3

C 5

D 7

E 9

Total 25

You are at a state fair and are offered a “Game of Chance.” You pay $1 to toss a dart at this dartboard.

Hit A: Get back $3, Hit B: Get back $2, Hit C: Get back $1Hit D or E: Lose $1

Is this a “FAIR GAME?“

Expected Value

1 3 5 16 6( ) ($2) ($1) ($1) ($1) $ $0.24

25 25 25 25 25E x

Label Label Probability

A 1/25

B 3/25

C 5/25

D 7/25

E 9/25