Alan

Let M be the mass. Then

M = 100 + M/2

Subtract M/2 from both sides

M/2 = 100

Multiply both sides by 2

M = 200 kg  (Heavy man!)

.

What is mcm?

.

a.  E(x) = (1/2)*(-1) + (1/2)*(1) = 0

b. var(x) = ((-1 - 0)² + (1 - 0)²)/2 = 1

c. P(|x - E(x)|>1) = 0,     P(|x - E(x)|=1) =   1

d.  See c.

Starting at 1 there is probability 1/2 that it moves to 0 in 1 step.  There is probability 1/2² that it moves to 0 in 3 steps (1 → 2 → 1 → 0).  There is probability 1/ 2³ that it moves to 0 in 5 steps (1 → 2 → 1 → 2 → 1 → 0), etc.  

The expected number of steps is therefore t₁ = 1/2 + 3/2² +  5/2³ + ...  or 

$$t_1=\sum_{k=1}^\infty\frac{2k-1}{2^k}=3$$

Starting at 2 it takes 1 step to 1 from where the expected number of steps is 3 as obtained above, so t₂ = 1+3 or t₂ = 4

So the ordered pair is (t₁, t₂) =  (3, 4)

.

The circumference is just pi*diameter, so if the diameter is 20ft the circumference is 20pi ft.

.

You can't take the square root of a negative value (not while dealing with real numbers, anyway), so what is the first integer bigger than pi?

Here's a picture to help:

.

I just used MathCad to work out your expression Melody.  

There is 1 ace of spades and there are 12 face cards.  The other 39 cards are irrelevant!  So this reduces to the probability of choosing the ace of spades first out of a pack of 13.  This probability is just 1/13.

As it happens, Melody, your complicated expression evaluates to 1/13.

. 

There are 22 possible ways of ending up where he started after 6 steps.  11 of them where his first step is in a clockwise direction and another 11 where his first step is in an anti-clockwise direction.  The direction of each step is chosen with probability 1/2, so overall probability = 22*(1/2)⁶ ≈ 0.34375

If we number the vertices as 0 1 2 3 4 5, where 0 is the initial vertex, then the following list shows the 11 routes from 0 back to 0 in which the first step goes to vertex 1 (the other 11 are obtained by interchanging the 1s and 5s and interchanging the 2s and 4s).

.

Expected value = (sum of weight*number)/(sum of weights)

So, for part 1:  Expected value = (6*1 + 2*3)/(6 + 2)  =  12/8  =  3/2

Part 2:  Expected value =  (6*1 + 3*3)/(6 + 3) = 15/9 = 5/3

 See if you can now do the other parts.

.