Spike is not a terribly bright student. His chances of passing chemistry are 0.35; mathematics, 0.40; and both, 0.12. Are the events "Spike passes chemistry" and "Spike passes mathematics" independent? What is the probability that he fails both subjects?

The events "Spike passes chemistry" and "Spike passes mathematics" are not independent.

The probability that he fails both subjects = 0.37

Step-by-step explanation:

The probability of Spike passing Chemistry = P(Chemistry) = 0.35

The probability of Spike passing Mathematics = P(Mathematics) = 0.40

The probability of Spike passing both Chemistry and Mathematics = P(Chemistry,Mathematics) = 0.12

For the events "Spike passes chemistry" and "Spike passes mathematics" to be independent, P(Chemistry,Mathematics) should be equal to P(Chemistry) * P(Mathematics)

But P(Chemistry) * P(Mathematics)=0.35*0.40 =0.14

So the two events are not independent.

The Probability that he passes Chemistry or Mathematics is given by P(Chemistry) + P(Mathematics) - P(Chemistry,Mathematics)

= 0.35+0.40-0.12=0.63

So the probability that he fails both subjects is the complement of this, namely (1 - 0.63) = 0.37

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!

answer: it's a

step-by-step explanation: