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

• mr_coffee
In summary, the problem states that a gambler plays successive games of blackjack until he loses 3 times in a row. The number of ways the gambler can play n games is denoted as g_n. The possible sequences for g_3 are WWW and LLL, for g_4 is WLLL, and for g_5 are LWLLL and WWLLL. The recurrence relation for g_n is a_k = a_{k-1} + a_{k-2} + a_{k-3} where k ≥ 6.
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!

Last edited:

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!

## 1. How do I know if I am hitting all possible cases when losing 3 times in a row?

In order to ensure that you are hitting all possible cases when losing 3 times in a row, you need to carefully analyze your data and the conditions under which the losses occur. This will help you determine if there are any patterns or factors that may be causing the losses and if there are any additional scenarios that need to be considered.

## 2. Are there any exceptions to the rule of being eliminated after 3 consecutive losses?

As with any scientific study or experiment, there may be exceptions or outliers that do not fit the expected pattern. It is important to carefully consider any potential outliers and determine if they should be included in the analysis or if they should be considered separately.

## 3. Is there a specific method or formula for determining all possible cases?

There is no one-size-fits-all method or formula for determining all possible cases. It will depend on the specific scenario and data being analyzed. However, there are various techniques and tools that can be used, such as probability analysis and decision trees, to help identify and consider all potential cases.

## 4. What is the significance of the "3 times in a row" rule?

The "3 times in a row" rule is often used in scientific studies as a threshold or cutoff point for determining the likelihood or significance of an event or outcome. In the case of losing 3 times in a row, it may indicate a trend or pattern that needs to be further investigated.

## 5. Can losing 3 times in a row be influenced by external factors?

While it is possible for external factors to influence the outcome of any experiment or study, it is important to control for these factors as much as possible. This may involve conducting the experiment in a controlled environment or adjusting the analysis to account for any potential external influences.

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