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Discrete Distributions (Alatissa expert in probability theory)

Re: Discrete Distributions (Alatissa expert in probability theory)

5 months 2 weeks ago - 5 months 2 weeks ago
#73
The turnover is considered very unexpectedly. Thank you.
The game is not gambling if you know how to win...

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Re: Discrete Distributions (Alatissa expert in probability theory)

5 months 2 weeks ago - 5 months 2 weeks ago
#74
Intuitively, it seems that the greater the loss for the selected bet, the more grounds there are for it to be realized on the current spin. No one argues with the probability of 1/37, but the chances that the bet will work seem higher.
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Re: Discrete Distributions (Alatissa expert in probability theory)

5 months 2 weeks ago - 5 months 2 weeks ago
#75
The next strategy is to wait for the entry point into the game.

2. If I want to bet 4 moves in a row in the neighbors of number "13" for $1, but before that its neighbors did not play 10 moves in a row, then what is the probability of winning?
...

A group of events through a geometric distribution would look like this:
p
qp
qqp
qqqp
qqqqp
qqqqqp
qqqqqqp
qqqqqqqp
qqqqqqqqp
qqqqqqqqqp
qqqqqqqqqqp
qqqqqqqqqqqp
qqqqqqqqqqqp
qqqqqqqqqqqqqp + qqqqqqqqqqqqqq

Complete group of events
p + qp + q2p + q3p + q4p + q5p + q6p + q7p + q8p + q9p + q10p + q11p + q12p + q13p +q14

By rates:

The player will virtually bet $0 each spin until there are ten losses in a row.
Next he will bet $5, if he loses another $5, if he loses another $5, after thirteen losses in a row he will either win or lose, but his $20 allocated for the session will be gone.

Mathematical expectation = financial result / turnover

Final result = p * 0 + qp *0 + q2p*0 + q3p * 0 + q4p * 0+ q5p * 0 + q6p * 0 + q7p * 0 + q8p * 0 + q9p * 0 + q10p * 31 + q11p (-5 + 31) + q12p * ( -5 -5 + 31) + q13p * ( -5 -5 -5 +31) + q14 * (-5 -5 -5 -5) =
= q10p * 31 + q11p * 26 + q12p * 21 + q13p * 16 + q14 * (-20) =
= (32/37)10 * 5/37 * 31 + (32/37)11 * 5/37 * 26 + (32/37) 12 * 5/37 * 21 + (32/37)13 * 5/37 * 16 + (32/37)14 * (-20) = - 0.10314244922...

Turnover = q10p * 5 + q11p * (5 + 5) + q12p * (5 + 5 +5) + q13p * (5+ 5 + 5 +5) + q14 * (5 + 5 + 5 +5) =
= (32/37)10 * 5/37 * 5 + (32/37)11 * 5/37 * 10 + (32/37) 12 * 5/37 * 15 + (32/37)13 * 5/37 * 20 + (32/37)14 * 20 =
= 3.81627062114...

Mathematical expectation = - 0.10314244922 / 3.81627062114 = - 0.027027...

As you can see, waiting a long time to enter the game (after consecutive losses) does not affect the mathematical expectation.
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Re: Discrete Distributions (Alatissa expert in probability theory)

5 months 2 weeks ago - 5 months 2 weeks ago
#76
mous writes:
Okay. I have $20 on my balance.
1. If I want to bet 4 moves in a row on the neighboring "13" numbers for $1, what is the probability of winning?
2. If I want to bet 4 moves in a row on the neighbors of number "13" for $1, but before that its neighbors did not play 10 moves in a row, what is the probability of winning?
3. If I want to place 4 moves in a row in the neighbors of the "13" number for $1, but before that its neighbors won 3 times in the previous 10 spins, then what is the probability of winning?
Here's the thing.
The player will double the bet if he loses.
plays four spins.
It is possible to calculate whether such bets will affect the mathematical expectation.

That is, he bets $5, and if the first one loses, he bets $10.
if both the first and second lose, the bet is $20
after the third loss of $40

Final result (of one spin) in case of winning:
in the first 31$
in the second 62$
in the third 124$
in the fourth 248$

A group of events through geometric distribution will look like this:
p
qp
qqp
qqqp + qqqq

Full group of events:
p + qp + q2p + q3p + q4

Then the mathematical expectation is
MO = financial results / turnover

Fin. result = р* 31 + qp * (-5 +62) + q2p* (-5 -10+ 124) + q3p* (-5 -10 -20 + 248) + q4 * (-5 -10 -20 -40) =
5/37* 31 + 32/37 * 5/37 * 57 + (32/37)2 * 5/37 * 109 + (32/37)3 * 5/37 * 213 + (32/37)4 * (-75) =
- 1.47256559...

Turnover = p * 5 + qp * (5+ 10) + q2p * (5 + 10 + 20) + q3p * (5 + 10 + 20 +40) + q4 * (5 + 10 + 20 + 40) =
5/37 * 5 + 32/37 * 5/37 * 15 + (32/37)2 * 5/37 * 35 + (32/37)3 * 5/37 * 75 + (32/37)4 * 75 = 54.48492685527..

Mathematical expectation = - 1.47256559 / 54.48492685527 = - 0.027027...

As you can see, the increase in rates did not affect the mathematical expectation.

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Re: Discrete Distributions (Alatissa expert in probability theory)

5 months 2 weeks ago - 5 months 2 weeks ago
#77
From the book Alice in Mathsland:
......







Alice's goal here is to find the sum of an infinitely decreasing geometric progression.

The numerical sequence b1, b2, b3, b4 ....
each member of which is equal to the previous one multiplied by a constant number q for this sequence, is called a geometric progression.
The number q is called the denominator of the progression. If the denominator |q|
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Re: Discrete Distributions (Alatissa expert in probability theory)

5 months 2 weeks ago - 5 months 2 weeks ago
#78
I will continue.
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