A peculiar die has the following properties

Question

A peculiar die has the following properties: on any roll the probability of rolling either a , a , or a is , just as it is with an ordinary die. Moreover, the probability of rolling either a , a , or a is again . However, the probability of rolling a is , not as one would expect of an ordinary fair die.

From what you know about this peculiar die, compute

1. the probability of rolling either a , a , or a ;
2. the probability of rolling neither a nor a .

Answer 1

The probability of rolling a 4 or 5 is 1/2 (probability of rolling 4, 5, or 1)  minus 3/8 (probability of rolling 1) = 1/8.

 

For a similar reason, the probability of rolling a 6 or a 2 is also 1/2 - 3/8 = 1/8

 

So the probability of rolling a 4, 5, 6 or 2 is 1/8 + 1/8 = 1/4.

 

We already know that the probability of rolling a 1 is 3/8.

 

So the probability of rolling a 1, 4, 5, 6 or 2 is 1/4 + 3/8 = 5/8

 

The only other side is a 3, so the probability of rolling that is what's left: 1 - 5/8 = 3/8.

 

Now we know following probabilities:

 

1:  3/8

2 or 6: 1/8

3:  3/8

4 or 5: 1/8

 

So the probability of rolling 3, 6, or 2 is 1/8 + 3/8 = 1/2

 

And the probability of not rolling 6 or 2 is 1 - 1/8 = 7/8