A 4-sided fair die is independently rolled two times

Question

A 4-sided fair die is independently rolled two times.

a) Define two events = “at least one of the rolls is 3” and = “at least one of the rolls is 2”. What are their probabilities?

b) Given has occurred, what is the probability of ? (Actually P(E|D)=?)

c) Given one of the rolls is a 3, what is the probability that the other one is a 2?

What is the difference between question b and question c?

Answer 1

a) To find D, first find the probability none of the rolls are $3$, then take $1$ minus that.

So, $D = 1 - (\frac{3}{4})^2 = \boxed{\frac{7}{16}}$

Similarly, E can be calculated in the same way and we can then find that

$E = \boxed{\frac{7}{16}}$

b) There are 2 cases here;

1. Only 1 roll of 3 has happened.

2. 2 rolls of 3 have happened.

Of the two cases, we see that D has a $\frac{3}{4}$ chance of being case 1 and a $\frac{1}{4}$ chance of being case 2 assuming that D has to happen first before E.

Then, only case 1 works, meaning that the other roll has to be 2 for E to work which has a $\frac{1}{4}$ chance of happening. There is only one possible case of this; Roll a $3$ then a $2$. This means that the answer to problem b is $\frac{1}{4}*\frac{3}{4} = \boxed{\frac{3}{16}}$.

c) Now we don't care about order in this problem. There are $2$ possible rolls for which one could be the $3$; the first roll or the second one. Either way, since rolling a 2 is an independent event, there is just a $\boxed{\frac{1}{4}}$ chance of this happening.

The difference between problem b and c is that problem c didn't care about order and rolling a 2 was an independent event. Problem b, however, did care about order, and rolling E was not a solely independent event.

Hope this helped you.