- #1

mr_coffee

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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, [tex]a_k = a_{k-1} + a_{k-2} + a_{k-3}[/tex] ?

Thanks!

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, [tex]a_k = a_{k-1} + a_{k-2} + a_{k-3}[/tex] ?

Thanks!

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