Rom

\(\text{so much text...}\\ A = \{\text{even on 2nd run}\}\\ B = \{\text{odd on 1st run}\}\\ \text{Choices}\\ P[A|B] = P[A]\\ P[A|B] = \dfrac{P[B\cap A]}{P[B]}\\ P[A|B] = \dfrac{P[B]}{P[A]}\\ P[A|B] = \dfrac{P[A]}{P[B]}\)

1) is true if A and B are independent

2) is always true

3) may or may not be true depending on A and B, it being true doesn't say anything about independence of A and B

4) if A and B are independent then P[A|B] = P[A], and thus this choice shows A and B are not independent

d**n you're right....

time for new glasses.

why did you switch it to an alternating series?

\(a=\dfrac 1 2\\ b = \dfrac{\sqrt{35}}{2}\)

xxxx

\(f^{-1}(g(x)) = x^3-1\\ g(x) = f(x^3-1)\\ x = g^{-1}(f(x^3-1))\\ 7 = 2^3-1\\ g^{-1}(f(7)) = g^{-1}(f(2^3-1)) = 2\)

\(A=32x^2+8x = \ell \cdot w\\ A = 8x(4x+1)\\ \text{Given the info you've provided there's really no further we can go}\\ \ell = 8x \\ w = (4x+1)\ \text{all you can really say about }x \text{ is }\\ x>0 \text{ since length must be positive}\)

The conditional probability of Event B given Event A is P(B|A)=P(A and B)/P(A) when two events are not independent.

I got 433 as well.

I think the confusion stems from whether a digit can be repeated or not. 

CPhil looks to be assuming they cannot.

There's nothing in the problem that states they can't.

The full list of numbers is

{200, 202, 201, 209, 220, 222, 221, 229, 210, 212, 211, 219, 290, 
292, 291, 299, 100, 102, 101, 109, 120, 122, 121, 129, 110, 112, 111, 
119, 190, 192, 191, 199, 900, 902, 901, 909, 920, 922, 921, 929, 910, 
912, 911, 919, 990, 992, 991, 999}

and this has mean 433

this curve produces the following points

(0,1), (2, 4)

the average rate of change between these two points is just the slope of the line connecting them.

\(m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{4-1}{2-0} = \dfrac 3 2\)