The Revised Case Study #1.10: Rare Event Approach

Case Description: Demonstrate the Rare Event Approach

Pairs of Dice and Rare Sums

Consider an experiment in which we toss a pair of fair, independently operating K-sided dice, and observe the sum of the face values.

Then we know the following:

For each individual die, if the die has K distinct, equally likely face values, then the probability for each distinct face value is P=(1/K).

In the pair-tossing experiment, there are the K*K equally-likely pairs, each with probability (1/K²), obtained as:

Pr{(d1,d2) shows} =

Pr{d1 shows from 1^st K-sided die}* Pr{d1 shows from 2^nd K-sided die} =

(1/K)*(1/K) =

1/( K²).

Moreover, if each K-sided die has face-values {1,2,3,…,K} then the event “SUM = 2” has probability (1/K²), obtained as:

Pr{Die Pair yields “SUM = 2”} = Pr{ (1,1) shows } = 1/(K²).

Case Objectives

Compute Pr{SUM = 2”} for the following cases: K=3, K=4, K=6, K=8 K=10, K=12, K=20 and K=30.

Compute the minimum sample size for each case, using the formula

n_”SUM=2” ≈ (1/ P_”SUM=2”)

We’ll explore these cases with samples of 100 tosses of selected pairs of dice.