Probability theory and mathematical statistics:

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1 N.I. Lobachevsky State University of Nizhni Novgorod Probability theory and mathematical statistics: Geometric probability Practice Associate Professor A.V. Zorine Geometric probability Practice 1 / 7

2 A point is chosen at random from the square below. What s the probability the point belongs to the red circle? Geometric probability Practice 2 / 7

3 A point is chosen at random from the square below. What s the probability the point belongs to the red circle? Let 2a be the side of the square, an elementary outcome is a point inside the square, Ω = G is the whole square, A = g is the circle. Area of G is (2a) 2, area of g is πa 2, P(A) = πa2 4a 2 = π 4 Geometric probability Practice 2 / 7

4 A point is chosen at random from a square. Square s side equals 2a. What s the probability that the distance between the point and the nearest side is less than a 3 (event A)? What s the probability that the distance between the point and the nearest side is less than x R 1 (event B x )? Geometric probability Practice 3 / 7

5 A point is chosen at random from a square. Square s side equals 2a. What s the probability that the distance between the point and the nearest side is less than a 3 (event A)? What s the probability that the distance between the point and the nearest side is less than x R 1 (event B x )? a 3 2a Ω = G is the square, A = g is the blue strip a 3 wide, area of G is 4a2, area of g is 4a 2 ( 2a 2 a ) 2 3 = 20a 2 9, P(A) = 5 9 Geometric probability Practice 3 / 7

6 A point is chosen at random from a square. Square s side equals 2a. What s the probability that the distance between the point and the nearest side is less than a 3 (event A)? What s the probability that the distance between the point and the nearest side is less than x R 1 (event B x )? If x 0 then B x = and P(B x ) = 0. If x > 0 and x is not too big so that we can use previous solution, then area of g is 4a 2 (2a 2x) 2, ( P(B x ) = 1 1 x ) 2 a This formula is true when x a. When x a, B x = Ω and P(B x ) = 1. Geometric probability Practice 3 / 7

7 A coin falls on a checked paper. Check s size is d, coin s radius is r (2r < d). What s the probability the coin falls clearly inside a check? Geometric probability Practice 4 / 7

8 A coin falls on a checked paper. Check s size is d, coin s radius is r (2r < d). What s the probability the coin falls clearly inside a check? The coins is clearly inside the check if its center is inside the yellow square. r d P(A) = (d 2r)2 d 2 Geometric probability Practice 4 / 7

9 Ann and Bart have a date tonight. They are to meet each other between 8 p. m. and 9 p. m. Ann waits for 10 minutes for Bart and Bart waits for 20 minutes for Ann. What s the probability they ll meet? Geometric probability Practice 5 / 7

10 Ann and Bart have a date tonight. They are to meet each other between 8 p. m. and 9 p. m. Ann waits for 10 minutes for Bart and Bart waits for 20 minutes for Ann. What s the probability they ll meet? 9 00 p.m. t B Ω = G = {(t A, t B ): 8 00 t A 9 00, 8 00 t B 9 00 }, { A = g = (t A, t B ): 8 00 t A 9 00, 8 00 t B 9 00, t A t B t A + 10 or } t B t A t B + 20, 8 00 p.m p.m. t A 9 00 p.m. area of g = , area of G = 60 2, P(A) = = Geometric probability Practice 5 / 7

11 A stick of length l is broken at two places randomly chosen. With what probability three pieces can make a triangle? Geometric probability Practice 6 / 7

12 A stick of length l is broken at two places randomly chosen. With what probability three pieces can make a triangle? From geometry, segments a, b and c can make a triangle if and only of sum of any two is greater than the third: a + b > c, a + c > b, and b + c > a. O Q R O 1 Let OQ = x, OR = y, 0 x l, 0 y l. x < y x + (y x) > l y, x + (l y) > y x, (y x) + (l y) > x Geometric probability Practice 6 / 7

13 A stick of length l is broken at two places randomly chosen. With what probability three pieces can make a triangle? From geometry, segments a, b and c can make a triangle if and only of sum of any two is greater than the third: a + b > c, a + c > b, and b + c > a. O Q R O 1 Let OQ = x, OR = y, 0 x l, 0 y l. x < y y > l/2, y x < l/2, x < l/2 Geometric probability Practice 6 / 7

14 A stick of length l is broken at two places randomly chosen. With what probability three pieces can make a triangle? From geometry, segments a, b and c can make a triangle if and only of sum of any two is greater than the third: a + b > c, a + c > b, and b + c > a. O R Q O 1 Let OQ = x, OR = y, 0 x l, 0 y l. x < y y > l/2, y x < l/2, x < l/2 x > y y + (x y) > l x, y + (l x) > x y, (x y) + (l x) > y Geometric probability Practice 6 / 7

15 A stick of length l is broken at two places randomly chosen. With what probability three pieces can make a triangle? From geometry, segments a, b and c can make a triangle if and only of sum of any two is greater than the third: a + b > c, a + c > b, and b + c > a. O R Q O 1 Let OQ = x, OR = y, 0 x l, 0 y l. x < y y > l/2, y x < l/2, x < l/2 x > y x > l/2, x y < l/2, y < l/2 Geometric probability Practice 6 / 7

16 y l l/2 Area of G is l 2, area of g is l2 4, P(A) = 1 4 O l/2 l x Geometric probability Practice 7 / 7

17 Real numbers p, q are chosen at random between 0 and 1. What s the probability equation x 2 + px + q = 0 has real roots? Real roots exist when p 2 4q 0. q 1 g = {(p, q): q p2 } 4 Area of g is O 1 q = p2 4 p 1 0 p 2 4 dp = 1 12 P(A) = 1 12 Geometric probability Practice 8 / 7

CN#5 Objectives 5/11/ I will be able to describe the effect on perimeter and area when one or more dimensions of a figure are changed.

CN#5 Objectives 5/11/ I will be able to describe the effect on perimeter and area when one or more dimensions of a figure are changed. CN#5 Objectives I will be able to describe the effect on perimeter and area when one or more dimensions of a figure are changed. When the dimensions of a figure are changed proportionally, the figure will