We have six dice and play the following game, which consists of two parts.
In the first part we determine the number N of dice to be used in the second part.
In the second part we roll N dice and take the sum of points S shown by the N dice.
We can see N and S as random variables and we can describe outcomes of a round by the values of N and S. Determine the following probabilities:
(i) Determine P[S = 2].
(ii) What is the probability that 1 die was rolled given that the sum shown is 2? That is, determine P[N=1|S =2].
Hint: Remember Bayes’ Law. (iii) What is the probability that the sum is 2, given that an even number of dice was rolled? That is, determine P[S =2|N is even].
(iv) What is the probability that an even number of dice was rolled, given that the sum is 2? That is, determine P[N is even|S =2].
I have the following ideas: the number of possibilities to obtain a given sum depends on the number of dice being rolled. For the question i, I can use the low of total probability but I do not know how to apply it since the number of dice is non defined. For example, I can have a sum of 2 by having 2 dice, each with a probability of 1/6 to land on 1. Therefore the total probability will be 1/36. Or by having one die landing on 2 with probability 1/6. But I do not know how to count the total probability given the 6 dice. Maybe you can help me clarify my ideas.
For the other questions I can use the Bayes' Law, but also here I have the same problem of considering the number of dice.
Thank you for your help!