Discrete Distributions (Alatissa expert in probability theory)

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

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

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

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

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

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.