Area Probability
Math 374
Game Plan
 Simple Areas
 Heron’s Formula
 Circles
 Hitting the Shaded
 Without Numbers
 Expectations
Simple Areas
 Rectangles
l
w
A=lxw
Always 2
A = Area,
l = length,
w = width
Trapazoid
Where A = Area
a
h
b
A = ½ h (a + b)
h = height
between parallel
line
a + b = the
length of the
parallel lines
Parallelogram
h
b
A=bxh
where A = Area
Triangles
Where A = Area
h
h = height
b = base
b
A = ½ bh or bh
2
Triangle Notes
Identify b & h
1
b
h
4 h
b
3
2 h
b
h
b
Simple Area
 Using a formula – 3 lines (at least)
 Eg Find the area
A = lw
8m
A = (12) (8)
12m
A = 96 m2
Simple Area
 Find the area
A = ½ bh
15m
A = ½ (20)(15)
A = 150 m2
20m
Simple Area
 Find the Area
8m
A = lw + (½ bh)
A = (9)(8)+((½)(3)(9))
A = 85.5 m2
9m
11m
Using Hero’s to find Area of
Triangle
 Now a totally different approach was found
by Hero or Heron
 His approach is based on perimeter of a
triangle
Be My Hero and Find the Area
 Consider
P = a + b + c (perimeter)
p = (a + b + c) / 2 or
p = P / 2 (semi perimeter)
b A = p (p-a) (p-b) (p-c)
a
c
Hence, by knowing
the sides of a
triangle, you can find
the area
Be My Hero and Find the Area
 Eg
P = 9 + 11 + 8 = 28
p = 14
A = p (p-a) (p-b) (p-c)
9
11 A = 14(14-9)(14-11)(14-8)
A = 14 (5) (3) (6)
8
A = 1260
A = 35.5
Be My Hero and Find the Area
P = 42 + 43 + 47
p = 66
A = p (p-a) (p-b) (p-c)
 Eg
42
43 A = 66(24)(23)(19)
47
A = 692208
A = 831.99
Be My Hero and Find the Area
P=9+7+3
p = 9.5
A = p (p-a) (p-b) (p-c)
 Eg
9
7
3
A = 9.5(0.5)(2.5)(6.5)
A = 77.19
A = 8.79
Do Stencil
#1 & #2 
Circles
d= 2r
r=½d
A = IIr2
d
r
d= diameter
r= radius
A = area
Circles
 In the world of mathematics you
always hit the dart boardA shaded = lw
 P (shaded) = A shaded A shaded = 16x16
A total
A shaded = 256
A Total = IIr2
10
A Total=3.14(10)2
A total=314
16
P = 256/314
P= 0.82
Probability Without Numbers
 Certain shapes are easy to calculate
 Eg. Find the probability of hitting the
shaded region
Expectation
 We need to look at the concept of a
game where you can win or lose and
betting is involved.
 Winning – The amount you get minus
the amount you paid
 Losses – The amount that leaves
your pocket to the house
Expectations
 Eg. Little Billy bets $10 on a horse that





wins. He is paid $17.
Winnings?
17 – 10 = $7
Expectation is what you would expect to
make an average at a game
Negative – mean on average you lose
Zero – means the game is fair
Positive means on average you win
Expectation
 In a game you have winning events and






losing events. Let us consider
G1, G2, G3 be winning events
W1, W2, W3 are the winnings
P, P, P are the probability
B1, B2 be losing events
L1, L2 be the losses
P (L1) P (L2) are the probability
Example
$12 B1
$5 G1
$3 G2
$10 B2
$2 G3
Loss
B1  L1 = $12
(P(L1) = 1/5
B2  L2 = $10
(P(L2) = 1/5
You win if you
hit the shaded
Win
G1  W1 = $5
(P(W1) = 1/5
G2  W2 = $3
(P(W2) = 1/5
G3  W3 = $2
(P(W3) = 1/5
Example Solution
 E (Expectancy) = Win – Loss
 = (W1 x (P(W1) + (W2 x (P(W2))
+ (W3 x (P(W3)) - (L1 x (P(L1)) +
(L2 x (P(L2))
= ((5 x (1/5) + 3 x (1/5) + 2 x (1/5)) –
((12 x (1/5) + 10 x (1/5))
= (5 + 3 + 2)
- ( 12 + 10)
5
5
Solution Con’t
 = 10 - 22
5
5
 -12/5 (-2.4) expect to lose!