# Show that the probability that he will never stops gambling is zero

• MHB
• Akea
In summary, we have shown that the probability of the gambler never stopping gambling is zero, and the probability of him having 500 dollars at the time of stopping is very small while the probability of having zero dollars is 0.5.
Akea
Can u help me with this question pls

Assume that a gambler plays a fair game where he can win or lose 1 dollar in each round . His initial stock is 200 dollar. He decides a priory to stop gambling at the moment when he either has 500 dollars or 0 dollars in his stock. Time is counted by the number of rounds played.
i) show that the probability that he will never stops gambling is zero
ii ) Compute the probability that at the time when he stops gambling he has 500 dollars and the probability that he has zero dollars

.

Sure, I can help you with this question. Let's break it down into two parts.

i) To show that the probability that the gambler will never stop gambling is zero, we can use the concept of infinite geometric series. In this case, the gambler's stock can either increase by 1 dollar or decrease by 1 dollar in each round, with equal probability. Therefore, the probability of him winning 1 dollar in a round is 1/2 and the probability of him losing 1 dollar is also 1/2.

Now, let's consider the probability that he will never stop gambling. This means that he will always have a stock of either 200 dollars or more. This can be represented as an infinite geometric series with a starting value of 200 dollars, a common ratio of 1/2 and an infinite number of terms. The sum of this series can be calculated as:

S = a / (1-r) = 200 / (1-1/2) = 400

Since this sum is finite, it means that the probability that the gambler will never stop gambling is zero.

ii) To compute the probability that the gambler has 500 dollars at the time when he stops gambling, we can use the concept of binomial distribution. The gambler can reach 500 dollars in two ways: either by winning 300 rounds (since he starts with 200 dollars) or by losing 200 rounds. The probability of winning 300 rounds out of 500 is given by:

P = (500 choose 300) * (1/2)^300 * (1/2)^200 = 0.000003

Similarly, the probability of losing 200 rounds out of 500 is also 0.000003. Therefore, the total probability of him having 500 dollars at the time of stopping gambling is 0.000006, which is a very small value.

On the other hand, the probability of him having zero dollars at the time of stopping gambling is 0.5, since he can either win or lose in each round with equal probability.

I hope this helps! Let me know if you have any further questions.

## 1. What does it mean to "never stop gambling"?

The phrase "never stop gambling" refers to a person continuously participating in gambling activities without ever taking a break or stopping.

## 2. How can someone prove that the probability of never stopping gambling is zero?

Statistically, it is impossible for a person to continuously gamble without ever stopping. This is because gambling involves random chance, and eventually, a person will either run out of money or reach a point where they no longer want to continue gambling. Therefore, the probability of never stopping gambling is zero.

## 3. Is there any scientific evidence to support this claim?

Yes, there have been numerous studies on gambling behavior and addiction that have shown that individuals who engage in continuous gambling often experience negative consequences and eventually stop gambling. Additionally, mathematical probability also supports the idea that the probability of never stopping gambling is zero.

## 4. Can someone become addicted to gambling even if they never stop?

Yes, someone can still develop a gambling addiction even if they never stop gambling. Addiction is often characterized by the inability to control one's behavior despite negative consequences, and this can happen even if a person never stops gambling.

## 5. Is there any way to prevent someone from never stopping gambling?

There are various measures that can be taken to help someone who may be at risk of continuously gambling. These include seeking professional help, setting limits and boundaries, and finding healthier alternatives to gambling. However, ultimately, it is up to the individual to make the decision to stop gambling.

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