If you lose 3 times in a row your out, am i hitting all cases?

[mr_coffee]

Hello everyone.

I"m not sure if I'm hitting all the cases here, the probem states:

A gambler decides to play successive games of blackjack until he loses 3 times in a row. (thus the gambler could play 5 games by losing the first, winning the second, and loosing the final three or by winning hte first two and losing hte final three. These possibilities can be symolized as LWLLL and WWLLL.) Let g_n be the number of ways the gambler can play n games.

a. find g_3, g_4, and g_5.

for g_3 = WWW, LLL
because if you only want to play 3 games, u can't say WLW, because then you would have to play an additional 3 games, to get out. (if I'm understanding this correctly).
But now that i just typed that, if he Wins all 3 games, then he isn't out. so would it only be LLL? if he only wants to play 3 games, he would have to looose all 3, if he won all 3, then he would keep going until he lost 3 in a row right?
g_4 = WLLL
g_5 = LWLLL, WWLLL

b. find g_6
g_6 = WWWLLL, LWWLLL, LLWLLL, WLWLLL

c. Find a reucrrence relation for g_3,g_4,g_5...

I won't be able to figure this out until i figure the beginning out but the hint is:

if k >= 6, any sequence of k games must begin with W, LW, or LLW, where L stands for "loose" and W stands for "win"

would it be, $$a_k = a_{k-1} + a_{k-2} + a_{k-3}$$ ?

Thanks!

 

[mighty2000]

Hello,

I am a scientist and I would like to offer some clarification on the problem and provide some solutions.

Firstly, for g_3, the possible sequences are WWW and LLL. This is because the gambler must lose 3 times in a row, so if they win all 3 games, they would not stop playing. Therefore, the only way for the gambler to stop after 3 games is to lose all 3 games.

For g_4, the only possible sequence is WLLL. This is because the gambler must lose 3 times in a row and then win the final game.

For g_5, the possible sequences are LWLLL and WWLLL. This is because the gambler could either lose the first game and then win the next 2 before losing 3 in a row, or win the first 2 games and then lose 3 in a row.

For g_6, the possible sequences are WWWLLL, LWWLLL, LLWLLL, and WLWLLL. This is because the gambler could win the first 3 games and then lose 3 in a row, or win the first 2 games, lose the third, and then lose 3 in a row, or lose the first game, win the next 2, and then lose 3 in a row, or win the first game, lose the next, win the third, and then lose 3 in a row.

For the recurrence relation, it would be a_k = a_{k-1} + a_{k-2} + a_{k-3} where k ≥ 6. This is because for any sequence of k games, the last game must be a win and the previous games can be any combination of the last 3 games (which is why we have a_{k-1}, a_{k-2}, and a_{k-3}).

I hope this helps and please let me know if you have any further questions. Good luck with your problem-solving!