The probability that jane will go to a ballgame (event a) on a monday is 0.73, and the probability that kate will go to a ballgame (event b) the same day is 0.61. the probability that kate and jane both go to the ballgame on monday is 0.52. from the given scenario, we can conclude that events a and b are . nextreset

Events A and B are called independent, when

otherwise events A and B are dependent.

The events A, B and A∩B are:

events A and B are dependent

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

8121 because

130,000 divided by 16 is 8121