Question

How many times of “3” are expected to get in rolling an unloaded dice 50 times?

Select one:

a. 25 times

b. 8 times

c. 16.67 times

d. 8.33 times

Answer 1

Let be a sequence of independent random variables that are all distributed according to a discrete uniform distribution over the numbers 1 to 6. These represent our dice rolls.

Now let be another sequence of variables defined as   That is, the indicators that tell us if the i’th die roll comesout 3 or not.

Now we have that and that   This implies that these Y-variables are in fact Bernoulli distributed with success parameter 1/6. So each Y is 1 if it’s corresponding die roll yielded 3 and 0 if it didn’t.

Now we can make yet another variable that counts the number of ‘successes’, that is, how many times we get the number 3 in our 50 die rolls. Thus, let
. Now it holds that this sum is binomially distributed with success parameter 1/6 and number of trials equal to 50.

Hence, we may now calculate the expected number of successes (i.e. threes) according to the formula for the expectation of the binomial distribution. That is, times .

So we see that the number of times we would expect the die to land with 3 up is indeed 8.33 out of 50.

option (d) is correct