Is the answer wrong? (from textbook)

1. The problem statement, all variables and given/known data

The question below, i already have the answer. but I can't work it out. Can anyone help?

Questions (Original)

In a game, Tom throws a fair dice twice. 1 point is awarded for each '6' thrown, 2 points are awarded if the sum of the two numbers thrown is prime, and no points are awarded for other outcomes. Find the probabilities that

(a) Tom gets 1 point in the game.
(b) Tom gets 2 points in the game.
(c) Tom gets at least 2 points in the game.


My attempts are as follows:

(a) There are total 36 possible results for a fair dice to be thrown twice. For each '6' thrown 1 point is awarded so there are (1,6) (2,6) (3,6) (4,6) (5,6) (6,6) and (6,1) (6,2) (6,3) (6,4) (6,5), but (6,6) is not counted because it gives two points, and (1,6) and (5,6) are also left out because they give prime number on sum.

So the desired outcomes should be (2,6)(3,6)(4,6)(6,4)(6,3)(6,2), then 6/36 = 1/6 but the answer is wrong. I am not going to say the answer but I just don't understand.

Please also explain (b) and (c) if possible. Thank you.

a) The only he gets one point is with 6 on 1st dice and 2,3 or 4 on the 2nd dice or the other way round.

First probability is 1/6x3/6=1/12 Second probabilty 3/6x1/6=1/12 So answer = 2/12 = 1/6

This is effectively the same as you did so I suspect the book is wrong.