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!