Design a betting system on a 7 series game beetween two teams A and B (even odds at each game), such as if A wins the series your cumulated gain is exacly £100 and if A looses your cumulated loss is exactly £100.
Had a shot at this, Basically, the trick is to work backwards. Firstly, consider a series with only 1 game, build to 3 games, then to 5 and finally to 7 1 Game Series If you start with no money, you simply bet $100 on A and you obtain the desired result. If you start with some money (X, negative or positive), you can never obtain desired result: Bet 0: X, X (need 100, -100) Bet X: 2X, 0 Bet 2X: 3X, -X Important result (1) 3 game series Starting with no money, make a bet of X on A. Due to symmetry, we will consider only the situation if A wins (B is skew symmetric, i.e. identical bit negative). A:B = 1:0 If A wins again, they have won the series hence they need to win a 100 total. However, if B wins, we are back to a 1 Game Series, and need to enter this stage with no money too. Hence, the bet is identical to the first bet, i.e X. If A wins, you have 2X = 100, X=50 and if A loses you have a 1 game series, following above strategy. Summary, bet $50 on A on game 1 and $50 on A on game 2. If they win both, you're up a $100, they lose both, you're down $100 and they draw, you have zero and it's a 1 game series. If you start with some money for a game 3 series, symmetry no longer holds, as after stage 1, you don't have skew symmetry. Say, you have X if A wins and -Y if B wins, in the event of a tie, you need to have 0 for a 1 game series to lead to desired result. However, if A wins game 2, you have 2X and if B wins you have -2Y, which means for 2X=100=2Y, we can't start with money. This argument holds for 5 and 7. 5 game series: Similarly as before, bet X on A and then X again on A. 1:1 Now a 3 game series with 0 start, recurse 2:0 You have 2X (or -2X in the converse) Now if A wins we need to bet the amount such that the end total is $100. If A loses, we have a 2:1, or a 1:0 lead in a 3 game series. Assume you bet Y 3:0 You have 2X+Y 2:1 You have 2X-Y Following the 2:1 result, which is identical to a 1:0 lead in a 3 game series, we look back and note that 2X-Y = 50 for the same stage in the 3 game series. And 2X+Y=100 for the winning total to be correct: 2X-Y = 50 2X+Y = 100 4X = 150 X = 75/2 Y=25 Summary: Bet 75/2 on A first 2 games. If it's a tie, follow 3 game series. If you win both (or lose both), bet 25 on the next round. If you win, you have a 100, if you lose you're at the 1:0 stage of the 3 game with 50 which can give desired outcome. 7 Game Series: Again as before, bet twice on A with X 1:1 - follow 5 game series 2:0: Bet Y on A (You have 2X in hand) If Lose the game: is a 2:1 result, or a 1:0 lead in a 5 game series You have 2X-Y in hand, and at the same stage in a 5 game series, you need 75/2 3:0: Now bet Z on A (2X+Y in hand) If you win, 2X+Y+Z = 100 You lose, you have 3:1 which is identical to the 2:0 lead in the 5 game series. From above, you need to have 75 at this stage (2X+Y-Z=75) Solving, using the last two eqn alone 2Z = 25 Z = 25/2 Everything else 2X+Y = 175/2 2X-Y = 75/2 4X = 250/2 X = 125/4 Y = 25 In summary, bet 125/4 on A for the first 2 games. If tie, follow 5 game series. Otherwise you've won both (or lost both). Then bet 25 on the next game. If you lose [2:1], follow 5 game series with 1:0 lead. Otherwise, bet 25/2 on the final game. If you win, you have the 100, otherwise [3:1] follow the 5 game series with a 2:0 lead.
I got the same answer as Varun. But I think it's easier to do in terms of a tree. Start with (0,0) and then two possible nodes (0,1) and (1,0). If either team reaches a 4, it is a terminal state, with value -100 or +100. You'll end up with a tree that widens and then narrows to (3,4) and (4,3). Now use backward induction. Label each node with Value|Bet.
You immediately bet 100$ that A will win the series and - that's it.
Do you want me to get your CV across to the CEO? I know him personally and we are on good terms.
Lol, salty are we? They do not need to test you for 'good' fit considering the review you wrote. Doubt most people would work with you. The level of arrogance because you have a MSc LOL. Most PhDs fail prop interviews.