[LVHerbie]

A friend and I have a bet where we both pick three games a week against the spread throughout the football season ("our season" ends the last week of regular college football so we have seven weeks, or 21 picks, left)... Right now he is up on me by 4 games and offered me a buy out of 2/3rds the orginal bet... My first impression was that it seemed like a good price but I thought I would post here to see if anyone had additional thoughts before I took him up on his offer...

[Ganchrow]

OK. I'll make assume the following assumptions:

1. Neither you nor your opponent may pick the same games.
2. Each of you has a 50% probability of winning any given pick.
3. Your pick outcomes are independent of that of your opponents.
4. Your bet with your opponent is at even odds.
5. A tie at the end of the season results either in no action or a coin-flip to determine the winner.
6. Winning picks are always worth the same unit, losing picks are worth the same negative unit, ties are not possible.

You're asking for the probability that over 21 picks you'll get exactly 4 more picks correct than your opponent (for a tie), and the probability that over those picks you'll get 5 or more more picks correct thatn your opponent.

So for the 1st part (end-of-season tie) here's a chart with the number of correct opponent picks in the first column, his probability of getting the specified number of picks correct in the second column, the number of picks you would need to get correct in order to tie him (i.e., get exactly 4 more correct than he) in the third column, your probability of getting the specified number of picks correct in the fourth column, and finally the probability of your opponent getting the specified number of picks correct AND your getting exactly the number of picks necessary to tie him in the fifth column.

You and Opponent Tie


# Opp.	|	Opp.		|	# Your	|	Your		|	Prob of
Picks	|	Prob		|	Picks	|	Prob		|	Both
----------------------------------------------------------------------------------------
21	|	 0.00005%	|		|	        	|	
20	|	 0.00100%	|		|	                |	
19	|	 0.01001%	|		|	        	|	
18	|	 0.06342%	|		|	        	|	
17	|	 0.28539%	|	21	|	 0.00005%	|	0.00000%
16	|	 0.97032%	|	20	|	 0.00100%	|	0.00001%
15	|	 2.58751%	|	19	|	 0.01001%	|	0.00026%
14	|	 5.54466%	|	18	|	 0.06342%	|	0.00352%
13	|	 9.70316%	|	17	|	 0.28539%	|	0.02769%
12	|	14.01567%	|	16	|	 0.97032%	|	0.13600%
11	|	16.81881%	|	15	|	 2.58751%	|	0.43519%
10	|	16.81881%	|	14	|	 5.54466%	|	0.93255%
9	|	14.01567%	|	13	|	 9.70316%	|	1.35996%
8	|	 9.70316%	|	12	|	14.01567%	|	1.35996%
7	|	 5.54466%	|	11	|	16.81881%	|	0.93255%
6	|	 2.58751%	|	10	|	16.81881%	|	0.43519%
5	|	 0.97032%	|	 9	|	14.01567%	|	0.13600%
4	|	 0.28539%	|	 8	|	 9.70316%	|	0.02769%
3	|	 0.06342%	|	 7	|	 5.54466%	|	0.00352%
2	|	 0.01001%	|	 6	|	 2.58751%	|	0.00026%
1	|	 0.00100%	|	 5	|	 0.97032%	|	0.00001%
0	|	 0.00005%	|	 4	|	 0.28539%	|	0.00000%
----------------------------------------------------------------------------------------
SUM	|								|	5.79034%

So what we see by summing up the fifth column (which for each is the product of the cells in the corresponding second and fourth columns), is that you have a 5.79034% probability of tying your opponent given the rules specified above.

Now for the 2nd part (end-of-season win for you) here's a chart with the number of correct opponent picks in the first column, his probability of getting the specified number of picks correct in the second column, the number of picks you would need to get correct in order to beat him (i.e.., get exactly 5 or more more correct than he) in the third column, your probability of getting that specified number of picks correct in the fourth column, and finally the probability of your opponent getting the specified number of picks correct AND your getting exactly the number of picks necessary to beat him in the fifth column.

You beat Opponent


# Opp. 	|	Opp.		|	# Your 	|	Your		|	Prob of
Picks	|	Prob		|	Picks	|	Prob		|	Both
----------------------------------------------------------------------------------------
21	|	 0.00005%	|		|	        	|	
20	|	 0.00100%	|		|	                |	
19	|	 0.01001%	|		|		        |	
18	|	 0.06342%	|		|	        	|	
17	|	 0.28539%	|		|	        	|	
16	|	 0.97032%	|	21	|	 0.00005%	|	0.00000%
15	|	 2.58751%	|	20	|	 0.00105%	|	0.00003%
14	|	 5.54466%	|	19	|	 0.01106%	|	0.00061%
13	|	 9.70316%	|	18	|	 0.07448%	|	0.00723%
12	|	14.01567%	|	17	|	 0.35987%	|	0.05044%
11	|	16.81881%	|	16	|	 1.33018%	|	0.22372%
10	|	16.81881%	|	15	|	 3.91769%	|	0.65891%
9	|	14.01567%	|	14	|	 9.46236%	|	1.32621%
8	|	 9.70316%	|	13	|	19.16552%	|	1.85966%
7	|	 5.54466%	|	12	|	33.18119%	|	1.83979%
6	|	 2.58751%	|	11	|	50.00000%	|	1.29375%
5	|	 0.97032%	|	10	|	66.81881%	|	0.64835%
4	|	 0.28539%	|	 9	|	80.83448%	|	0.23069%
3	|	 0.06342%	|	 8	|	90.53764%	|	0.05742%
2	|	 0.01001%	|	 7	|	96.08231%	|	0.00962%
1	|	 0.00100%	|	 6	|	98.66982%	|	0.00099%
0	|	 0.00005%	|	 5	|	99.64013%	|	0.00005%
----------------------------------------------------------------------------------------
SUM	|								|	8.20747%

So what we see here is that your probability of beating your opponent would be 8.20747%, and hence your probability of losing would be 1-5.79034%-8.20747% ≈ 86.00219%.

So assuming you and your opponent each wagered a unit, then your current EV would be about 8.20747% - 86.00219% ≈ -77.79472%. Hence your settling for a payment of 75% of your initial bet would be a good move for you.

[Ganchrow]

For both charts, the Excel formula in the 2^nd column for row X would be: =BINOMDIST(AX, 21, 0.5, 0), which is assuming the first column of the excel chart starts at Excel column A.

For the first chart, the Excel formula in the 4^th column for row X would be: =BINOMDIST(CX, 21 ,0.5, 0), which is assuming the first column of the excel chart starts at Excel column A.

For the second chart, the Excel formula in the 4^th column for row X would be: =1-BINOMDIST(CX-1, 21, 0.5, 1), which is assuming the first column of the excel chart starts at Excel column A.

[sportsbo]

mmm, all that above seems pretty complicated. I say, you are a gambler right, go for it. win it all or lose it all!! after all it is just friendly entertainment right, not a pro handicapping contest in vegas...