Task 3: Probabilities of interest in two-dice experiments

Likelihood of two events:

• A = {at least one 6 when rolling a single dice 4 times}
• B = {at least one pair 66 when rolling two dice 24 times}

The probability of a   6   ⁣in a single roll of a dice is 1/6, so in a long sequence of four rolls, we may expect to see a   ⁣6   ⁣about 4×(1/6) = 2/3 of time.

Likewise, the probability of a pair   ⁣66   ⁣in a single roll of two dice is 1/36; hence, in a long sequence of 24 rolls of two dice, we may expect to see   ⁣66   ⁣about 24×(1/36) = 24/36 = 2/3 of time – the same!

From this consideration, it may seem that events A and B are equally likely.

Is that correct? The subtlety of the question is in the words “at least one”!

The tasks for you are now the following:

• Design and conduct two separate simulations to try and evaluate the probabilities P(A) and P(B) numerically. 
  Note: Because these probabilities may be expected to be quite close to each other (if not identical), you will need to use quite a lot of replications (perhaps 100,000 or so).
• Evaluate the MOE in both simulations, which should provide sufficient confidence (e.g 95%) in discriminating between the two probabilities.