| |
BGonline.org Forums
Think about correlation between shots
Posted By: Bob Koca In Response To: Think about correlation between shots (Jeremy Bagai)
Date: Saturday, 27 May 2017, at 4:03 p.m.
"that the one with the greater chance of equaling 2 gives also the greater chance of equaling 0 which matches survival. Not understood."
Suppose you have a probability distribution with outcomes 0, 1, 2 and expected value 2/3. If you then adjust it to have a larger chance of equaling 2 you need to adjust probabilities for equaling 0 and or 1 so that the sum of probabilities is still 1. If you want the expected value to still equal 2/3 you cannot do this without increasing the chance of equaling 0. Given the value of P2 and the expected value the probability of equaling 0 can be solved for:
P0 + P1 + P2 = 1 (1)
P1 + 2P2 = 2/3 (2)
P1 = 2/3 - 2P2 from (2)
P0 + (2/3 - 2P2) + P2 = 1 Subbing into (1)
P0 = P2 + 1/3 Solving for P0
Thus if P2 is greater then P0 is also greater.
"Since f(x) = x^2 is a concave up function the variability increases that chance. Not understood"
In the first case P2 = (2/6)^2 and in second case P2 = 1/2(1/6)^2 + 1/2(3/6)^2.
The concavity gives a quick way to know that the second value is larger. Graph the function f(x) = x^2 and choose any two points on the curve. If you draw the line segment connecting those 2 points it lies above the curve. A linear interpolation gives a greater value than the value of the function.
This idea means the argument extends to cases such as choosing 4 bullets for sure or 1/2 chance of either 3 or 5.
| |
BGonline.org Forums is maintained by Stick with WebBBS 5.12.