thin blade
A player in roulette always bets on a color (black or red). If he wins, he gets double his stake back. He uses the Martingale strategy: In the first round, he starts with the minimum bet, e.g. 1€. If he loses, he doubles his stake in the next round. If he wins, he goes back to the minimum bet in the next round. What winnings or losses are possible after three rounds of play? What is the probability of getting a loss after three rounds?
heavy iris
@thin blade Hey, what did you try? Did you figure out some possible winnings or losses?
thin blade
<@749514204926705677> Hey, what did you try? Did you figure out some possible wi...
I don’t know how to figure that out but rn I’m trying to solve the second question using the bernoulli formula
p = 18/37
n = 3
x = 1
am I on the right track?
“What’s is the probability of getting a loss after three rounds”
heavy iris
I don’t know how to figure that out but rn I’m trying to solve the second questi...
Well in order to compute this probability, you first need to find the possible winnings/losings.
Can you be more precise on what you call the Bernoulli formula? It looks like you try to use the binomial distribution but this is not a problem where you can use it
thin blade
Well in order to compute this probability, you first need to find the possible w...
Oh yeah it’s the binomial distribution!!
How can I find the possible winnings/losings?
Is it combinatorics?
heavy iris
Well if you play three games, you can first determine the possible sequences of winning and losing a round, (hint: there are 8 of them)
For example, you can win the three rounds (WWW). Another example: you can lose the three rounds (LLL)
Can you find the 6 other outcomes?
thin blade
Well if you play three games, you can first determine the possible sequences of ...
WLL
WLW
WWL
LWW
LWL
LLW
heavy iris
Exactly, and now you can compute how much you will/lose for all these 8 outcomes
And this will be the answer to your first question
thin blade
Exactly, and now you can compute how much you will/lose for all these 8 outcomes
Does that mean I have to do ex.
WWW = 1 + 1 + 1 = 3
WLW = 1 + 2 + 1 = 4
Etc ?
sorry if I seem a lil slow 😭
nvm that doesn’t make sense does it
heavy iris
Don't worry
Not exactly, what you wrote is how much you put money on stake. Can you now compute the overall gain/loss?
heavy iris
nvm that doesn’t make sense does it
It makes half sense
thin blade
Don't worry
Not exactly, what you wrote is how much you put money on stake. Can ...
How do I do that? If it’s WLL would it be 1/3 (one win out of three)
heavy iris
Let's do one together
If you win all 3 rounds
You win 2 Euros on all 3 rounds, but you bet 1 Euro on all 3 rounds too, by definition of the game
It means you won 6 Euros after betting 3 Euros
Your gain overall is 6-3 = 3 Euros
thin blade
Ohhhh
heavy iris
Now, you have to do this for all the 7 other outcomes
thin blade
Then after calcuating those do I have to see how many were losses?
heavy iris
It's a bit long, but well, there's no better way. 😄
Exactly
For the second question
But right now you should focus on the first question
thin blade
You win 2 Euros on all 3 rounds, but you bet 1 Euro on all 3 rounds too, by defi...
I’m a lil confused, If I bet 1€, do I get my euro back or do I get an additional euro
heavy iris
It depends if you win or lose the round
thin blade
If I win oops
heavy iris
Basically if you bet x Euros, you lose it if you lose the round, and you win 2x Euros if you win the round
thin blade
ohhhh okay thank you!!
heavy iris
So, it's either you win x Euros, or either you lose x Euros, after a round
Because the overall gain when you win a round is 2x - x = x
-x comes from the initial bet
thin blade
WLL: I bet 7€ overall
on the first round I bet 1€ and I get +1
on the second -2
on the third -4
so overall -5€
I initially had 7€
I end up with 2€
Right?
heavy iris
If WLL happens, you bet: 1 + 1 + 2 overall, not 1+2+4
thin blade
omg ur right
heavy iris
Because after the first win, you go back to the minimum bet
And for the three rounds you win 2 + 0 + 0
Which means that your overall gain/loss is : 2 - 4 = -2
thin blade
WWW = +3€
WLL = -2€
WWL = +1€
WLW = +2€
LLL = -7€
LLW = +1€
LWL = 0€
LWW = +2€
How can I use these to solve the second question?
heavy iris
Alright, this looks good
To compute the probability of losing money, then you have to compute the probability that one of the two following outcomes happens : WLL or LLL
Because these are the only two cases in which you lose money
thin blade
To compute the probability of losing money, then you have to compute the probabi...
isn’t it 2/8?
heavy iris
Actually, no, because on a given round, you have more chance to lose than to win, because of that green 0
For one single round, your probability to win is 18/37, which is slightly less than 1/2
heavy iris
Are you familiar with the multiplication rule in probability?
Or even more simply you can use the independence between each rounds
For example: P(LLL) = P(L) * P(L) * P(L)
By independence of the outcomes for each rounds
So P(LLL) = (19/37)^3
thin blade
For example: P(LLL) = P(L) \* P(L) * P(L)
Is it right this way?
P(WLL) = P(W) * P(L) * P(L)
P(WLL) = (18/37) * (19/37) * (19/37 )
heavy iris
Yes
Finally, it remains to add those two results together
To find the probability of losing money
thin blade
WLL = 0.128
LLL= 0.135
= 0.2636
= 26,36%
Is that the probability?
I have a 26,36% chance of losing money?
heavy iris
Yes, this is right
empty rose