Dr. Pepper-
1) Derive the odds of 125,994,627,894,135 combinations by
2) there are 14 possible ratios of hearts to non-hearts: 0/20, 1/20, 2/20, 3/20, 4/20, 5/20, 6/20, 7/20, 8/20, 9/20, 10/20, 11/20, 12/20, 13/20
The following chart contains the rest of the work:
Hearts Of OddsH Odds NonHeart Total Odds Ratio * Total Odds
0 20 0 68,923,264,410 68,923,264,410 0
1 20 13 68,923,264,410 896,002,437,330 44,800,121,867
2 20 78 62,359,143,990 4,864,013,231,220 486,401,323,122
3 20 286 51,021,117,810 14,592,039,693,660 2,188,805,954,049
4 20 715 37,711,260,990 26,963,551,607,850 5,392,710,321,570
5 20 1,287 25,140,840,660 32,356,261,929,420 8,089,065,482,355
6 20 1,716 15,084,504,396 25,885,009,543,536 7,765,502,863,061
7 20 1,716 8,122,425,444 13,938,082,061,904 4,878,328,721,666
8 20 1,287 3,910,797,436 5,033,196,300,132 2,013,278,520,053
9 20 715 1,676,056,044 1,198,380,071,460 539,271,032,157
10 20 286 635,745,396 181,823,183,256 90,911,591,628
11 20 78 211,915,132 16,529,380,296 9,091,159,163
12 20 13 61,523,748 799,808,724 479,885,234
13 20 1 15,380,937 15,380,937 9,997,609
Total OddsH: 8,191 (not important)
Total Odds Non-Heart: 343,797,240,803 (not important)
Total Odds: 125,994,627,894,135 (very important)
Total (Odds * ratio), line-by-line, then totaled: 31,498,656,973,534 (very important)
Total odds next card is a heart 125,994,627,894,135 / 31,498,656,973,534 = 0.25 (this is the proof)
To demonstrate the calculation involved where there are 10 hearts and 10 non-hearts:
((13*12*11*10*9*8*7*6*5*4)/(1*2*3*4*5*6*7*8*9*10)) * ((39*38*37*36*35*34*33*32*31*30)/(1*2*3*4*5*6*7*8*9*10))
Hopefully the above columns that I tried to line up came out OK.
Now that I’ve schvitzed over this, what method would a normal person use to prove the 1/4 likelihood of the next card being a heart?