Objective- To solve problems involving probability and geometric shapes

If a dart hits the board below, what is the probability that it will land in the circle?

20

20

3

If a dart hits the board below, what is the probability that it will land in the circle?

20

20

3

P (circle) = favorablepossible

=Area of circleArea of square

r 2

s2

P (circle) (3.14)(3)2

(20)2 (3.14)(9)

4000.07065

P (circle) 7.1%

Points A, B, C, D, and E represent points onan interstate highway.

A B C D E

If a random accident occurs on AE, find the probability that it will occur between B and C.

P(accident is in BC) = BCAE

17 8 13 12

= 817 + 8 + 13 + 12

8P(accident is in BC) =

50= 0.16 = 16 %

Find the probability that the spinner will landon region D.

30

90

130 110

A B

C

D

P(D)=degrees in D

degrees in circle

P(D)= 110

360 0.305

P(D)= 30.5%

P(D)= 305

9%

Find the probability that a dart will land in the red area.

3

2 5

P(red area) =

r 2 r 2

r 2

middle - smalllarge

P(red area) =

(5)2 (3)2

(10)2P(red area) =

25 9100

P(red area) = 16100

16

10016%

P(red area) =red areatotal area

•If a parachutist jumps out of a plane, what is the probability he will land on the emblem of a Football Field?

Examples:Examples:

108 P(triangle)= 25

30sin1085.

%5.255.78

20

Examples:Examples:

Radius = 7 inRadius = 12 in

What is the probability of hitting the section of blue or anywhere inside the blue ring?

P=

P = 34%

2

2

12

7

Theoretical vs. Experimental Probability

• Theoretical probability – When all outcomes are equally likely, the probability that an event will occur is calculated by dividing the desired outcome by all possible outcomes.

• Experimental probability is calculated by performing an experiment or trial, recording the actual outcomes and dividing how many times the desired outcome actually happened by the number of trials performed.

Fundamental Counting Principle

• If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is m x n.

• This principle can be extended to three or more events.

• Police use photographs of various facial features to help witnesses identify suspects. One basic identification kit contains 195 hairlines, 99 eyes and eyebrows, 89 noses, 105 mouths and 74 chins and cheeks. How many different faces can be produced with this information?

• The standard configuration for a New York license plate is 3 digits followed by 3 letters. ( 257 KPD)

• -How many different license plates are possible if digits and letters can be repeated?

• -How many different license plates are possible if digits and letters cannot be repeated?

Permutations

• An ordering of n objects is a permutation

• For example, there are six permutations of the letters A, B, C

• ABC, BCA, CBA, ACB, BAC, CAB– You can use the fundamental counting principle

to determine the number of permutations of n objects, it is n! Or n x (n-1) x (n-2) …….

– Or n! (n factorial)

• Twelve skiers are competing in the final round of the Olympic freestyle skiing aerial competition.

• -In how many different ways can the skiers finish the competition?

• -In how many different ways can 3 of the skiers finish first, second and third to win a gold, silver or bronze medal?

Permutations of n objects taken r at a time

)!(

!

rn

nPrn

You are considering 7 different colleges. You want to visit allOr some of them. In how many ways can you visit 4 of them? All 7 of them?

Permutations with repetition

• Find the number of distinguishable permutations of the letters in MISSISSIPPI

There are 11 letters I is repeated 4 times, S is repeated

4 times and P is repeated 2 times

Therefore:

650,34!2!4!4

!11

Formula for permutations with repetition

• The number of distinguishable permutations of n objects where one object is repeated q times :

• Find the number of distinguishable permutations in the word MASSACHUSETTS

!!......!!

!

321 nqqqq

n