Explore Geometry: Area, Probability & Games

Area Probability Math 374

Game Plan • Simple Areas • Heron’s Formula • Circles • Hitting the Shaded • Without Numbers • Expectations

Simple Areas • Rectangles l w A = Area, l = length, w = width A = l x w Always 2

Trapazoid Where A = Area h = height between parallel line a + b = the length of the parallel lines a h b A = ½ h (a + b)

Parallelogram where A = Area h b A = b x h

Triangles Where A = Area h = height b = base h b A = ½ bh or bh 2

Triangle Notes Identify b & h 1 b h 4 h b 3 h h 2 b b

Simple Area • Using a formula – 3 lines (at least) • Eg Find the area A = lw A = (12) (8) A = 96 m2 8m 12m

Simple Area • Find the area A = ½ bh A = ½ (20)(15) A = 150 m2 15m 20m

Simple Area • Find the Area A = lw + (½ bh) 8m 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 P = a + b + c (perimeter) • Consider p = (a + b + c) / 2 or p = P / 2 (semi perimeter) a A = p (p-a) (p-b) (p-c) b Hence, by knowing the sides of a triangle, you can find the area c

Be My Hero and Find the Area P = 9 + 11 + 8 = 28 • Eg 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) A = 1260 8 A = 35.5

Be My Hero and Find the Area P = 42 + 43 + 47 • Eg p = 66 A = p (p-a) (p-b) (p-c) 42 43 A = 66(24)(23)(19) A = 692208 47 A = 831.99

Be My Hero and Find the Area P = 9 + 7 + 3 • Eg p = 9.5 A = p (p-a) (p-b) (p-c) 9 7 A = 9.5(0.5)(2.5)(6.5) Do Stencil #1 & #2  A = 77.19 3 A = 8.79

Circles d= 2r r = ½ d d A = IIr2 r d= diameter r= radius A = area

Circles A shaded = lw A shaded = 16x16 A shaded = 256 • In the world of mathematics you always hit the dart board • P (shaded) = A shaded A total A Total = IIr2 A Total=3.14(10)2 A total=314 10 P = 256/314 P= 0.82 16

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? • 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 17 – 10 = $7

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 You win if you hit the shaded $12 B1 $5 G1 Win $3 G2 G1 W1 = $5 (P(W1) = 1/5 $2 G3 $10 B2 Loss G2 W2 = $3 (P(W2) = 1/5 B1 L1 = $12 (P(L1) = 1/5 G3 W3 = $2 (P(W3) = 1/5 B2 L2 = $10 (P(L2) = 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!

Explore Geometry: Area, Probability & Games