Discrete Distributions (Alatissa expert in probability theory)

Open Excel



press f(x)

a dialog box opens

select the category "statistics"
and further BINOMDIST






The following dialog box opens



fill out



"the number of successes" is the number of times r
"number of tests" is the limiting interval n
"probability of success" - for number 1/37, for six 6/37, etc. (i.e. probability every spin)
"integral"
here you choose FALSE
this will be the probability density

That's it. The value of the desired probability will appear automatically.

I am amazed at how smart and bright your head is!!

Binomial distribution

I will show in detail on a segment of 10 spins
probability how many times for number

p=1/37 each spin

formula p^rn = Cⁿr * p^r * (1-p) ^nr
where Cⁿr is the binomial coefficient,
r - number of times
n - limiting segment

alternative entry bk
n! / r! (n - r)!

First, calculate the binomial coefficient.

10! / 0! 10! =1
10! / 1! 9! = 10
10! / 2! 8! = 45
10! / 3! 7! = 120
10! / 4! 6! = 210
etc.

Now the values in the formula

Never p(0) = 1 * (1/37) ⁰ * (36/37)¹⁰ ≈ 0.76034
once p(1) = 10 * (1/37)¹ * (36/37)⁹ ≈ 0.21121
two times p(2) = 45 * (1/37)² * (36/37)⁸ ≈ 0.02640
three times p(3) = 120 * (1/37)³ * (36/37)⁷ ≈ 0.00196
four times p(4) = 210 * (1/37)⁴ * (36/37)⁶ ≈ 0.0001
etc.

In percentage

p(0) = 76.034%
p(1) = 21.121%
p(2) = 2.64%
p(3) = 0.196%
p(4) = 0.01%
etc.

I'll go into Excel now and make screenshots of how to calculate all this automatically.

and as for combinations

Dear Alatissa!

Your invaluable knowledge of probability theory should be applied to roulette.
Very often you give ready-made results on probability density without chewing on the formulas you use: there is no step for specific tasks where to substitute what and how to derive the final numbers.

We want to make a universal interactive machine for calculating the probabilities of combinations entering the RNG.
Let's call it "Mathematical Expertise from Alatissa".

Initial data:
d - length of the segment (arbitrary value, specified by the user)
comb - combination ("17", "23-32", "7-17-27", "1-20-14-31-9-6-34-17", "27-16", etc., specified by the user)

Button "Calculate occurrences"

Result:

a) Number of occurrences in the specified segment - % probability
0- % 0.00217
1-% 0.00823
2-% 0.01974
3-% ...
4-%
5-%
6- ...
7-
8-
9-
10-

b) Average waiting time for the specified combination to appear.

The average time for your combination to appear is 124 spins.

* * *
To do this, we ask you to describe the entire calculation in detail using an example:

Length: 60 spins
Combination: "23-32"

.......................process

I see Shpilevoy is eager to become my co-host)

Discrete random variables are variables whose number of values can be counted.

* Number of hits on target with n shots. Accepted values 0…n
* The number of heads in n coin tosses. Accepted values are 0…n
* The number of points thrown when a dice is rolled. The random variable takes one of the values — {1,2,3,4,5,6}

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Probability distribution



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and then... troubles )))))

DISCRETE DISTRIBUTION - a probability distribution concentrated on a finite or countable set of points of a sample space Ω. More precisely, let ω₁, ω₂, ... be sample points and
p_i = р(ω_i), i = 1, 2, ..., (1)
are some numbers that satisfy the conditions
p_i ≥ 0, ∑_ip_i = 1. (2)
Relations (1) and (2) completely determine the differential equation in the space Ω, since the probability roulette of any set A ⊂ Ω is defined by the equality
P(A) = ∑_{i:ωi∈A}p_i.
Accordingly, the distribution of a random variable X(ω) is called discrete if with probability 1 it takes a finite or countable number of different values x_i with probabilities p_i = Ρ{ω : A(ω) = x_i}. For a D. r. on a straight line, the distribution function F(x) = ∑_{i:xi