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Area Probability Math 374

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Game Plan Simple Areas Simple Areas Herons Formula Herons Formula Circles Circles Hitting the Shaded Hitting the Shaded Without Numbers Without Numbers Expectations Expectations

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

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

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

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

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

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Simple Area Using a formula – 3 lines (at least) Using a formula – 3 lines (at least) Eg Find the area Eg Find the area 8m 12m A = lw A = (12) (8) A = 96 m 2

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Simple Area Find the area Find the area 20m 15m A = ½ bh A = ½ (20)(15) A = 150 m 2

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Simple Area Find the Area Find the Area 8m 9m 11m A = lw + (½ bh) A = (9)(8)+((½)(3)(9)) A = 85.5 m 2

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Using Heros to find Area of Triangle Now a totally different approach was found by Hero or Heron Now a totally different approach was found by Hero or Heron His approach is based on perimeter of a triangle His approach is based on perimeter of a triangle

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Be My Hero and Find the Area Consider Consider 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

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Eg Eg Be My Hero and Find the Area 9 8 11 P = 9 + 11 + 8 = 28 p = 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

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

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Eg Eg Be My Hero and Find the Area 9 3 7 P = 9 + 7 + 3 p = 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

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

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

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Probability Without Numbers Certain shapes are easy to calculate Certain shapes are easy to calculate Eg. Find the probability of hitting the shaded region Eg. Find the probability of hitting the shaded region

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Expectation We need to look at the concept of a game where you can win or lose and betting is involved. 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 Winning – The amount you get minus the amount you paid Losses – The amount that leaves your pocket to the house Losses – The amount that leaves your pocket to the house

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Expectations Eg. Little Billy bets $10 on a horse that wins. He is paid $17. Eg. Little Billy bets $10 on a horse that wins. He is paid $17. Winnings? Winnings? Expectation is what you would expect to make an average at a game Expectation is what you would expect to make an average at a game Negative – mean on average you lose Negative – mean on average you lose Zero – means the game is fair Zero – means the game is fair Positive means on average you win Positive means on average you win 17 – 10 = $7

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Expectation In a game you have winning events and losing events. Let us consider In a game you have winning events and losing events. Let us consider G 1, G 2, G 3 be winning events G 1, G 2, G 3 be winning events W 1, W 2, W 3 are the winnings W 1, W 2, W 3 are the winnings P, P, P are the probability P, P, P are the probability B 1, B 2 be losing events B 1, B 2 be losing events L 1, L 2 be the losses L 1, L 2 be the losses P (L 1 ) P (L 2 ) are the probability P (L 1 ) P (L 2 ) are the probability

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Example $5 G 1 $12 B 1 $2 G 3 $10 B 2 $3 G 2 You win if you hit the shaded G 1 W 1 = $5 (P(W 1 ) = 1/5 G 2 W 2 = $3 (P(W 2 ) = 1/5 G 3 W 3 = $2 (P(W 3 ) = 1/5 B 2 L 2 = $10 (P(L 2 ) = 1/5 B 1 L 1 = $12 (P(L 1 ) = 1/5 Win Loss

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Example Solution E (Expectancy) = Win – Loss E (Expectancy) = Win – Loss = (W 1 x (P(W 1 ) + = (W 1 x (P(W 1 ) + (W2 x (P(W 2 )) + (W 3 x (P(W 3 )) - (L 1 x (P(L 1 )) + (L 2 x (P(L 2 )) = ((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 5 5

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Solution Cont = 10 - 22 = 10 - 22 5 5 5 5 -12/5 (-2.4) expect to lose! -12/5 (-2.4) expect to lose!

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