Solutions to Practice Problems for  the Lander Math Contest
Module 13, Solutions (40-42)

40)
The probability of x being between 0.5 and 1 is equal to
the area of the region bounded by the probability density
function f(x) = - 2x + 2, the x-axis, the vertical line x=0.5 and
the vertical line x=1. [Draw a picture of this region in the
plane. Note: Our probability density function f(x) = - 2x + 2 is a
straight line with an x intercept at (1,0) and a y-intercept at
(0,2).] This region in the plane is a right triangle with horizontal
leg of length 0.5 and vertical leg of length 1. Since the area of
a triangle is equal to .5*b*h, the area is equal to (0.5*1)/2=0.
25 .

41) Let us apply the binomial formula P(k)= nC4 p^k(1-p)^n-k
for n=0,1,2,3,4,5 and 6. [Note that for all n, pk(1-p)n-k = 1/26]
P(0)=1*1/2^6 = 0.015625
P(1) =6*1/2^6 = 0.09375
P(2) =15*1/2^6 = 0.234375
P(3) =20*1/2^6 = 0.3125
P(4) =15*1/2^6 = 0.234375
P(5) =6*1/2^6 = 0.09375
P(6) =1*1/2^6 = 0.015625

42) For the unfair coin case we obtain (p=0.75):
P(0)=1*0.25^6 ≈ 0.0002
P(1) =6*0.75*0.25^5 ≈ 0.0044
P(2) =15*0.75^2*0.25^4 ≈ 0.0330
P(3) =20*0.75^3*0.25^3 ≈ 0.1318
P(4) =15*0.75^4*0.25^2 ≈ 0.2966
P(5) =6*0.75^5*0.25 ≈ 0.3550
P(6) =1*0.75^6 ≈ 0.1780