[RESOLVED] Probability calculations

[.paul.]

I’m working on a probability calculator, where coin flip probability for 3 heads from 3 flips is (1/2)^3*100

What I’m not sure how to calculate is from 4 flips, what is the probability of 2 heads and 2 tails?

After that I’ll be looking for how to calculate the probabilities when throwing regular 6 sided dice…

[Arnoutdv]

I would do with it with the negative binomial.

Head probability -> pH = 0.5, Tail -> pT = 0.5

/Edit: This is not correct
p = 1 - (1 - pH) * (1 - pH) * (1 - pT) * (1 - pT)

From Copilot:
Total possible outcomes:
Each flip has 2 outcomes (H or T), so for 4 flips:
Total outcomes = 2^4 = 16.


Number of favorable outcomes:
We want exactly 2 heads and 2 tails. The number of ways to arrange 2 heads among 4 flips is given by the binomial coefficient:
(4 over 2) -> 4! / (2! * 2) = 6
So there are 6 sequences like HHTT, HTHT, etc.


Probability:
Each sequence has probability:
(1/2)^4 = 1/16

Multiply by the number of favorable sequences:
Probability=6×(1/16) = 6/16 = 0.375

[Arnoutdv]

This is the Binomial Distribution.

P(X=k) = (n over k) * p^k * (1 - p)^(n-k)

Where
(n over k) = number of combinations (n choose k)
p = probability of success (for a fair coin, p=0.5)
n = number of trials
k = number of successes

[.paul.]

Thanks…

I understand most of that, except the favourable outcomes calculation. How would I scale that, for example 6 flips and a wanted outcome of 4 heads, 2 tails?

Edit: I was referring to post #2

[Arnoutdv]

The p would be: 1/6
Each roll has 6 possible outcomes, so for 4 rolls -> 6^4 = 1296 possible outcomes

A generated function:

Imports System

Module DiceProbability
    ' Function to calculate factorial
    Private Function Factorial(n As Integer) As Double
        Dim result As Double = 1
        For i As Integer = 1 To n
            result *= i
        Next
        Return result
    End Function

    ' Function to calculate n choose k (binomial coefficient)
    Private Function BinomialCoefficient(n As Integer, k As Integer) As Double
        Return Factorial(n) / (Factorial(k) * Factorial(n - k))
    End Function

    ' Function to calculate probability
    Public Function ProbabilityOfExactHits(nRolls As Integer, targetHits As Integer, sides As Integer) As Double
        Dim p As Double = 1.0 / sides           ' Probability of rolling the target number
        Dim q As Double = 1.0 - p              ' Probability of NOT rolling the target number

        Dim combinations As Double = BinomialCoefficient(nRolls, targetHits)
        Dim probability As Double = combinations * Math.Pow(p, targetHits) * Math.Pow(q, nRolls - targetHits)

        Return probability
    End Function

    Sub Main()
        Dim rolls As Integer = 4
        Dim hits As Integer = 2
        Dim sides As Integer = 6

        Dim result As Double = ProbabilityOfExactHits(rolls, hits, sides)
        Console.WriteLine($"Probability of getting {hits} times a 5 in {rolls} rolls: {result:P4}")
    End Sub
End Module

[.paul.]

Thanks. I’ll try that and see how I can use it. I understand all of post #2 except the binomial coefficient (I must’ve missed school that year )