Question

Roll a fair die 100 times and let T be the sum of the 100 results.

(a) Find a very good approximation for the value of P( T > 315 )

(b). Find P( T = 315 ), approximately.

Answer 1

For each dice throw, we have here:

P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = P(X = 5) = P(X = 6) = 1/6

Therefore, E(X) = (1/6)*(1 + 2 + 3 + 4 + 5 + 6) = 3.5

E(X²) = (1/6)*(1² + 2² + 3² + 4² + 5² + 6²) = 91/6

Var(X) = E(X²) - [E(X)]² = (91/6) - 3.5² = 2.9167

Therefore for a sum of 100 rolls, the distribution for the sum T could be given here as:

The probability here is computed as:

P(T > 315 )

Converting this to a standard normal variable, we get:

Getting it from the standard normal tables, we get:

Therefore 0.9798 is the required probability here.

b) The required probability here is:

P( T = 315 )

Applying the continuity correction factor, we get here:

Converting this to a standard normal variable, we get:

Getting this from the standard normal tables, we get here:

Therefore 0.0029 is the required probability here.