Quote:
Originally Posted by engineerEXECUTIVE
This is a classic argument in my opinion and I always like to bring up this point;
Lets say we were flipping a coin ten times.
For the first five flips to our surprise comes heads.
Now for the following flips ; 6,7,8,9,and finally 10.
Probability says the results are individual but theoretically if I bet
3x on tails the 6th bet , 9x on tails the seventh bet, and increasing the bet size(12x,25x,65x) on
tails each time heads wins; I should come out ahead.
(also we decided not to bet on the first five flips.)
I know I should not feed a T, but I had this from last year.
A quick change of values and copy/paste
Let us just think this through with some elementary math.
Given that we just saw HHHHH
That probability is = 1
It is a certain. It already happened.
Anything that has already happened
has a probability OF happening of exactly 1.
Code:
Flip Result Prob Prob Sequence
HHHHH 1 1 (it already happened)
6 H 0.5 0.5
The 6th flip still has a 50/50 prob of landing Heads
So Given we already have HHHHH with a prob of 1
and we get another head with a prob of 0.5
we multiply the 2 together. 1*0.5=0.5
The prob of getting 6 heads in a row,
given that we already had 5 in a row is exactly 50%
7 H 0.5 0.25
6 in a row = 0.5
For the 7th flip we still have a 50/50 chance of a Heads
So again we multiply the two to get 7 in a row.
0.5(the chance of getting 6 in a row)*0.5 = .25
So, a 25% chance of getting 7 Heads in a row
given we started out with 5 in a row.
The 8th, 9th and 10th flip follows the same method.
8 H 0.5 0.125
9 H 0.5 0.0625
10 H 0.5 0.03125********** this equals 1/32
So there is a 1/32 chance of getting 10 Heads in row,
given that we started out with 5 in a row.
For 10 in a row, from the very first flip,
we only have a 1/1024 chance.
Now to the math.
I used the unit betting sequence of 3, 9, 27, 81, 243 (I love 3s)
One really can use any betting sequence, just play with the numbers if you want.
Code:
Bet Prob of making the bet
3 1 (100% of the time we make a 3 unit bet)
9 0.5 and so on.
27 0.25
81 0.125
243 0.0625
363 total units LOST if heads hits 5 in a row.
That event only happens 1/32 of the time or 0.03125
The expected value of our loss, over playing this coin flip experiment many times,
-363*0.03125 = -11.34375
Now we need the expected win so the world can see the expected value of this betting system.
Code:
Net win Prob Return
3 0.5 1.5
6 0.25 1.5
15 0.125 1.875
42 0.0625 2.625
123 0.03125 3.84375
Add up the return column...
11.34375 is our expected win from this system.
-11.34375 + 11.34375 = 0 the expected value of playing this game.
Who knew? Raise their hand high so I can count you
This does not mean that at any time one can not be ahead.
It may take a long time and more bets to get ahead after one has fallen behind.
There is still variance.
But that can wait for another year.
As some have said earlier, the expectation does not change by the way the bets are made.
Sally