Area Probability Math 374. Game Plan Simple Areas Simple Areas Herons Formula Herons Formula Circles Circles Hitting the Shaded Hitting the Shaded Without.

Area ProbabilityArea ProbabilityMath 374Math 374

Game PlanGame Plan

Simple AreasSimple Areas Heron’s FormulaHeron’s Formula CirclesCircles Hitting the ShadedHitting the Shaded Without NumbersWithout Numbers ExpectationsExpectations

Simple AreasSimple Areas

Rectangles Rectangles l

w

A = l x w

Always 2

A = Area, l = length, w = width

TrapazoidTrapazoid

a

b

h

Where A = Area

h = height between parallel line

a + b = the length of the parallel linesA = ½ h (a + b)

ParallelogramParallelogram

h

b

where A = Area

A = b x h

Triangles Triangles

Where A = Area

h = height

b = baseb

h

A = ½ bh or bh

2

Triangle Notes Triangle Notes

hb

b

h

h

bb

h

1

2

4

3

Identify b & h

Simple AreaSimple Area

Using a formula – 3 lines (at least)Using a formula – 3 lines (at least) Eg Find the areaEg Find the area

8m

12m

A = lw

A = (12) (8)

A = 96 m2

Simple AreaSimple Area

Find the areaFind the area

20m

15m

A = ½ bh

A = ½ (20)(15)

A = 150 m2

Simple AreaSimple Area

Find the AreaFind the Area

8m

9m

11m

A = lw + (½ bh)

A = (9)(8)+((½)(3)(9)) A = 85.5 m2

Using Hero’s to find Area of Using Hero’s to find Area of TriangleTriangle

Now a totally different approach was found Now a totally different approach was found by Hero or Heronby Hero or Heron

His approach is based on perimeter of a His approach is based on perimeter of a triangletriangle

Be My Hero and Find the AreaBe My Hero and Find the Area

ConsiderConsider

a

c

b

P = a + b + c (perimeter)

p = (a + b + c) / 2 or

p = P / 2 (semi perimeter)

A = p (p-a) (p-b) (p-c)

Hence, by knowing the sides of a triangle, you can find the area

EgEg

Be My Hero and Find the AreaBe My Hero and Find the Area

9

8

11

P = 9 + 11 + 8 = 28p = 14

A = p (p-a) (p-b) (p-c)

A = 14(14-9)(14-11)(14-8)

A = 14 (5) (3) (6)

A = 1260

A = 35.5

EgEg

Be My Hero and Find the AreaBe My Hero and Find the Area

42

47

43

P = 42 + 43 + 47p = 66

A = p (p-a) (p-b) (p-c)

A = 66(24)(23)(19)

A = 692208

A = 831.99

EgEg

Be My Hero and Find the AreaBe My Hero and Find the Area

9

3

7

P = 9 + 7 + 3p = 9.5

A = p (p-a) (p-b) (p-c)

A = 9.5(0.5)(2.5)(6.5)

A = 77.19

A = 8.79

Do Stencil #1 & #2

CirclesCircles

d

d= diameterr= radius

r

d= 2r

r = ½ d

A = IIr2

A = area

CirclesCircles In the world of mathematics you In the world of mathematics you

always hit the dart board always hit the dart board P (shaded) = P (shaded) = A shadedA shaded

A totalA total

10

16

A shaded = lw

A shaded = 16x16

A shaded = 256

A Total = IIr2

A Total=3.14(10)2

A total=314 P = 256/314

P= 0.82

Probability Without NumbersProbability Without Numbers

Certain shapes are easy to calculateCertain shapes are easy to calculate Eg. Find the probability of hitting the Eg. Find the probability of hitting the

shaded regionshaded region

ExpectationExpectation

We need to look at the concept of a We need to look at the concept of a game where you can win or lose and game where you can win or lose and betting is involved. betting is involved.

Winning – The amount you get minus Winning – The amount you get minus the amount you paidthe amount you paid

Losses – The amount that leaves Losses – The amount that leaves your pocket to the houseyour pocket to the house

ExpectationsExpectations

Eg. Little Billy bets $10 on a horse that Eg. Little Billy bets $10 on a horse that wins. He is paid $17.wins. He is paid $17.

Winnings?Winnings? Expectation is what you would expect to Expectation is what you would expect to

make an average at a gamemake an average at a game Negative – mean on average you loseNegative – mean on average you lose Zero – means the game is fairZero – means the game is fair Positive means on average you winPositive means on average you win

17 – 10 = $7

ExpectationExpectation In a game you have winning events and In a game you have winning events and

losing events. Let us consider losing events. Let us consider

GG11, G, G22, G, G33 be winning events be winning events

WW11, W, W22, W, W33 are the winnings are the winnings

P, P, P are the probabilityP, P, P are the probability

BB11, B, B22 be losing events be losing events

LL11, L, L22 be the losses be the losses

P (LP (L11) P (L) P (L22) are the probability) are the probability

ExampleExample

$5 G1

$12 B1

$2 G3$10 B2

$3 G2

You win if you hit the shaded

G1 W1 = $5 (P(W1) = 1/5G2 W2 = $3 (P(W2) = 1/5G3 W3 = $2 (P(W3) = 1/5

B2 L2 = $10 (P(L2) = 1/5

B1 L1 = $12 (P(L1) = 1/5

Win

Loss

Example SolutionExample Solution E (Expectancy) = Win – LossE (Expectancy) = Win – Loss

== (W (W11 x (P(W x (P(W11) + ) + (W2 x (P(W(W2 x (P(W22))))

+ (W+ (W33 x (P(W x (P(W33)) - )) - (L(L11 x (P(L x (P(L11)) +)) +(L(L22 x (P(L x (P(L22))))

= ((5 x (1/5) + 3 x (1/5) + 2 x (1/5)) – = ((5 x (1/5) + 3 x (1/5) + 2 x (1/5)) – ((12 x (1/5) + 10 x (1/5)) ((12 x (1/5) + 10 x (1/5))

= (= (5 + 3 + 2)5 + 3 + 2) - ( - ( 12 + 1012 + 10))

5 55 5

Solution Con’tSolution Con’t

= = 1010 - - 2222

5 55 5 -12/5 (-2.4) expect to lose!-12/5 (-2.4) expect to lose!