How high is the probability that a hunter has an Unleash the Hounds combo in their hand?
Playing around the Starving Buzzard and Unleash the Hound combo is probably the most important factor in a match against a hunter. Letting them draw too many cards and clear the board with the hounds can easily lose you a game. But playing too few minions in fear of UtH can be equally dangerous, especially for decks that rely on having many minions on the board.
Until now I mostly went with gut feeling on the decision to play around UtH or limit the number of minions on my side. But I'd like to put my decisions on a more solid base here.
What is the actual probability that a hunter has Unleash the Hounds and Starving Buzzard on the hand at any point in time? It should be possible to calculate that, taking into account the mulligan phase, tracking and any other card draw. The result would obviously depend on the number of rounds played, or more exactly on the number of cards drawn by the hunter.
Best Answer
Since the combo consists of two different cards, but each card has two identical copies, it's easiest to break this down into a two-part question:
1) What is the chance that he has UTH?
2) Assuming that he has UTH, what is the chance that he has Buzzard?
Let x be the number of cards drawn in total. That means 30-x cards remain in the deck. The chance that neither copy of UTH is in his hand is: (30-x)*(29-x)/(30*29)
Therefore, the chance that he has at least one copy of UTH is: 1-((30-x)*(29-x)/(30*29))
Assuming that UTH is in his hand, there are 29 cards left that we care about, 29-x of which are still in the deck. The chance that neither copy of buzzard is in his hand is: (29-x)*(28-x)/(29*28)
Therefore, the chance that he has at least one copy of buzzard is: 1-((29-x)*(28-x)/(29*28))
Multiplying those two results together gives the answer.
Plugging in possible values for x:
4. 6.60%
5. 9.94%
6. 13.77%
7. 18.03%
8. 22.64%
9. 27.52%
10. 32.60%
11. 37.82%
12. 43.11%
13. 48.42%
14. 53.69%
15. 58.86%
16. 63.89%
17. 68.73%
18. 73.34%
19. 77.67%
20. 81.71%
21. 85.40%
22. 88.72%
23. 91.66%
24. 94.17%
25. 96.26%
26. 97.89%
27. 99.07%
28. 99.77%
29. 100.00%
30. 100.00%
Edit: correcting typo (I had referred to UTH instead of buzzard in the second equation)
Also, Jason has a valid point regarding mulligan affecting the percentage. However, it doesn't affect it in the ways implied so far.
In the simplest case throws back everything. Since the cards are being reshuffled into the deck before redraw, it is as if the opening draw never happened. Therefore, you can use the table above just as it is written.
Where the mulligan actually affects the percentages is if the hunter keeps cards. In the simplest case, assume the hunter keeps only UTH and Buzzard and throws back everything else. If he keeps 1, you know with 100% that he has either UTH or buzzard, and can use 1-((29-x)*(28-x)/(29*28)) to determine his chance of having the full combo. (I did not include the percentage for that but they can be easily computed) Note that x would include the card kept but not the cards thrown back.
The problem is that you don't know what algorithm the hunter uses. (Maybe he also keeps tracking, or hyena, or always keeps at least 1 card no matter what) Now it's like poker, where the raw percentages are a good starting point, but cannot be solely relied on. You have to balance probability with intuition.
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Answer 2
You want to calculate the Hypergeometric Distribution based on your wish for 2 different cards out of 4 out 30.
You can get the correct answer here: http://stattrek.com/online-calculator/hypergeometric.aspx
Answer 3
Deck = 30 cards.
Going first = 27 cards.
Draw to 3 mana(cost of UTH) = 24 cards.
Assuming 2 UTH cards in the deck = 2/24 which is 8% chance of them being able to cast UTH with no other buffs.
Needing 5 mana with starving buzzard = 4/22 cards 18% chance assuming they have 2 starving buzzards and 2 UTH in their deck.
If they use Tracking I guess you can do the math, but it gets more complicated. Obviously if they go second it's slightly different and the later in the game the higher the chance of this happening.
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