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

[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

 

[jvicens]

.

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.