Question

Consider a game in which ten true dice are rolled. Find the probability of obtaining

a. Exactly one “ace”

b. At least one “ace”

c. At most 4 aces.

d. What is the total number of microstates?

Answer 1

A game is considered, ten true dice are rolled.

Now, we are considering getting aces from this rolls.

Now, on each roll, an ace can be the outcome, with probability 1/6.

So, we can say that, if X be the random variable de noting the number of aces in 10 rolls, then X follows binomial with parameters n = 10 and p = 1/6.

So, the probability mass function of X is given by

where, i can take any integer value from 0 to 10.

question a

Probability of getting exactly one ace.

question b

Probability of getting at least one ace.

question c

Probability of getting at most 4 aces.

question d

What is the total number of microstates.

Now, the total number of microstates in an experiment is the number of elements in the sample space.

Now, for each roll of the die, there are 6 outcomes.

So, in the sample space, there are 6^10 elements.

So, the total number of microstates is