A standard six sided dice is tossed repeatedly. Let N be the total number of observed...

Question

Question

A standard six sided dice is tossed repeatedly. Let N be the total number of observed 1s and 2s.

For independent individual outcomes, calculate p(N=infinity)

(Hint continuity)

Full solution with justification.

Answer 1

Answer #1

When a die is thrown n times

P(N = k) =

P(k-0.5 <= N <= k+0.5) {continuity correction }

N follow binomial distribution with n and p = 1/3

N follow normal distribution with mean =n/3 and sd =sqrt(n/3 * 1/3*2/3) =sqrt(2n/27)

hence

P(N = k) =

P((k-0.5 - n/3)/sqrt(2n/27) < Z < (k+0.5 -n/3)/sqrt(2n/27))

when n tend to infinity and k tend to infinity

(k-0.5 - n/3)/sqrt(2n/27) (k+0.5 - n/3)/sqrt(2n/27)

hence

P(N = k) = 0

Answer 2

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