00:01
Hello, in this question it is being said that the gambler figures out that he can always beat the house by doubling his bet every time to recap any of his losses.
00:14
Okay, so he will quit as soon as he wins.
00:18
Otherwise, he will keep on doubling his vet up to the point when he will lost his funds to make his bet double.
00:28
Okay, so if he has one 50 bets, $150 .50.
00:32
So he has $150 to start with to have complete this is the money he have in total and he will start with one dollar then he has probability of winning one by two and losing one by two if he win he will quit if he lose he will again play with the problem with two dollar bet similarly again, he can either win or lose.
01:09
If he win, he will quit.
01:11
Or if he lose, he will continue playing.
01:14
So, similarly, if he lose, he will again double his bat.
01:20
That would be how much? $4.
01:25
This would be $4 .00.
01:28
Win or lose.
01:30
Sorry, this is half.
01:33
I made the mistake over here and over here.
01:38
This was probabilities written.
01:40
These were the probabilities return okay so he has probability half of winning and half of loging if he bet two dollars now similarly if he bet four dollars now he will either win or either lose similarly we will draw our table okay so now i will put this in the form of a table so as per the table i will make column one for his bet first he will bet one then he will bet two then he will bet four then eight and similarly so 16 32 64 if he still has the money okay so let me calculate how much money he will be having after he lose okay so probability of winning here is one by two in all of the cases it is 0 .5 only so let me write 0 .5 probability of winning every time so it would be 0 .5.
02:51
Sorry, priority of winning for the first bet is 0 .5.
02:57
And for the second bat, it would be something different.
03:01
Winning probability and losing probability.
03:04
Rosing probability is every time 0 .5.
03:06
This is 0 .5.
03:07
This is 0 .5.
03:08
This is 0 .5.
03:10
Okay, because he will only play the second bet if he lose the first one.
03:16
So, winning here is 0 .25 percentage chances.
03:22
Similarly, losing here is 0 .25 and winning here has a chance of, if he lose only then he will play the next one.
03:29
So here it has a chance of winning.
03:33
How much? let me calculate these values 1 by 1 .0 .25 into 0 .5.
03:40
That would be 0 .125.
03:45
Similarly, here also we will have 0 .125 percent chance.
03:51
Of winning similarly next one would be 0 .0 625 so 0 .0 625 winning 16 would be 0 .3125 0 .3125 let me once calculate up to what point he will have to bet or he can bet until unless he runs out of money so remaining amount i will write here because if he lose the first one he now only has $149 left he lose the second one now he has only $147 left now he has $143 left again he lose 8 now he has how much $135 left now if you lose 16 he has now $119 left now he lose 32 how much he has only how much $119 minus 32 that would be $87 he will have with him and now if he loses again $64 he will left with 23 and he can no longer double his back so that's it he will quit here so here so this value was 0 .30 -035 sorry okay similarly 0 .0 315 and the next one would be 0 .015625 similarly and next would be 0 .0078125 and same here.
05:53
So his expected value can be calculated by multiplying the winning amount into the probability.
05:59
Okay, so he can win either he can win one dollar with the probability of how much, 0 .5.
06:05
So it would be 1 into 0 .5, that would be 0 .5.
06:08
2 into 0 .25...