Question

suppose we take a die with 3 on three sides 2 on two sides and 1 on one side, roll it n times and let Xi be the number of times side i appeared find the conditional distribution P(X2=k|X3=m)

Answer 1

For any roll, probability of appearnce of number 3 , 2 and 1 are

P3 = 3/6 = 1/2

P2 = 2/6 = 1/3

P1 = 1/6

P(X2=k | X3=m) = P(X2 = k, X3 = m) / P(X3 = m)

In 'n' roll, 3 appeared m times, 2 appeared k times, then 1 appeard n - (m + k) times.

thus,

P(X2=k | X3=m) = P(X2 = k, X3 = m, X1 = n-k-m ) / P(X3 = m)

By multinomial distribution,

P(X2 = k, X3 = m, X1 = n-k-m ) = [ n! / (k! * m! * (n-k-m)!) ] * (1/2)^m * (1/3)^k * (1/6)^n-k-m

Now, if X3 appears m times, then any number other than 3 appears for n-m times.

and P3 = 1/2 and 1-P3 = 1 - (1/2) = 1/2

Using binomial distribution,

P(X3 = m)  = ⁿC_m * (1/2)^m * (1/2)^n-m = [n! / (m! (n-m)!] (1/2)^m * (1/2)^n-m

So,

P(X2=k | X3=m) = P(X2 = k, X3 = m, X1 = n-k-m ) / P(X3 = m)

= { [ n! / (k! * m! * (n-k-m)!) ] * (1/2)^m * (1/3)^k * (1/6)^n-k-m } / { [n! / (m! (n-m)!] (1/2)^m * (1/2)^n-m }

= [(n-m)! / (k! * (n-k-m)!) ] [ (1/3)^k * (1/6)^n-k-m / (1/2)^n-m ]

= ^n-mC_k * 2^n-m / (3^k * (2 * 3)^n-k-m )

= ^n-mC_k * 2^k / 3^n-m