Examples

Binomial estimation: 10,000 flips

Flip a fair coin 10,000 times. Write for the number of heads.

Estimate the probability that .

Solution

(1) Check the rule of thumb: and , so and the approximation is effective.

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(2) Now, calculate needed quantities:

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(3) Set up CDF:

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(4) Compute desired probability:

Summing 1000 dice

About 1,000 dice are rolled.

Estimate the probability that the total sum of rolled numbers is more than 3,600.

Solution

(1) Let be the number rolled on the die.

Let , so counts the total sum of rolled numbers.

We seek .

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(2) Now, calculate needed quantities:

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(3) Set up CDF:

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(4) Compute desired probability:

Continuity correction of absurd normal approximation

Let denote the number of sixes rolled after rolls of a fair die. Estimate .

Solution

We have , and and .

The usual approximation, since is continuous, gives an estimate of 0, which is useless.

Now using the continuity correction:

The exact solution is 0.0318, so this estimate is quite good: the error is 1.9%.