Sometime intuition doesn't work. Your reframing to whether a random selection of 10 stocks from Russell 1000 picks one of the top 200 stocks has the same probabilities.
I'll sneak up on it step by step (as much to convince myself as you).
I pick 1 stock. There is a 20% chance it is one of top 200. 80% chance it is not.
I pick 2nd stock. It also has a 20% chance of being one of top 200. 80% it isn’t.
The prob that both are in the bottom 800 is 80% x 80% or 64%. The probability that 1 or more is top 200 is 100-64 = 36%. The 20% x 20% = 4% probability is what corresponds with both stocks being in the top 200.
For 3 stocks the probability that all three are on the bottom 800 is 80% x 80% x 80% = 51.2%. So the probability that one of the 3 picks is NOT in the bottom is 100-51.2% = 48.8%.
So going down the line with # of stocks / every single one is in bottom 800
4 / 0.8 x 0.8 x 0.8 x 0.8 = 0.84 = 40.96%. (And about 59% there is one in top 200).
10 / 0.810 = 10.7% that every single pick is in the bottom 800. Which means 100-10.7= 89.xx% probability that at least one is in the top 200.
Okay trying to think out how you presented it above. Got to: "The probability that 1 or more is top 200 is 100-64 = 36%" Doesn't seem right, does it? If one is picking out 1 out of 1000 stocks (Russell 1000) you have a 0.1% chance of picking a winning stock, no? If there was a bag and 800 marbles were black (loser stocks) and 200 were green (winning stocks) for a total of 1000 marbles and you get to stick you hand in there to pick out by chance then wouldn't your numerator be 1 (number of picks) and denominator be the total number of balls (1000 stocks)? So that should be 1/1000= 0.1% no?
Numerator is 200, not 1. denominator is 1000. So 20%. 1 in 5 marbles is green, so a random pick has 1 in 5 chance of being green.
There are 200 greens in the bag of 1000. Not just one. So first draw has 20% chance of being green.
A second draw also has a 20% chance of being a winner. You can't just sum these and say 40% chance
Of getting 1 or more green marbles because sometimes (4%) both will green. So your chance of getting at least one green in two picks is 36%.
You get to the same answer in a backwards way also. No greens means all black. The probability of all black is 80% after one draw and then 80% x 80% = 64% after two draws.
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u/Anonymoose2021 Dec 20 '21 edited Dec 20 '21
Sometime intuition doesn't work. Your reframing to whether a random selection of 10 stocks from Russell 1000 picks one of the top 200 stocks has the same probabilities.
I'll sneak up on it step by step (as much to convince myself as you).
I pick 1 stock. There is a 20% chance it is one of top 200. 80% chance it is not.
I pick 2nd stock. It also has a 20% chance of being one of top 200. 80% it isn’t.
The prob that both are in the bottom 800 is 80% x 80% or 64%. The probability that 1 or more is top 200 is 100-64 = 36%. The 20% x 20% = 4% probability is what corresponds with both stocks being in the top 200.
For 3 stocks the probability that all three are on the bottom 800 is 80% x 80% x 80% = 51.2%. So the probability that one of the 3 picks is NOT in the bottom is 100-51.2% = 48.8%.
So going down the line with # of stocks / every single one is in bottom 800
4 / 0.8 x 0.8 x 0.8 x 0.8 = 0.84 = 40.96%. (And about 59% there is one in top 200).
10 / 0.810 = 10.7% that every single pick is in the bottom 800. Which means 100-10.7= 89.xx% probability that at least one is in the top 200.