2. a dice is rolled. a card is drawn from a deck of face cards (i. e. only jack, queen, king of each suit). a coin is flipped. a) draw a tree diagram to define the sample space for this game. b) you win the game if you toss tails, draw a heart, and roll an even number. find the probability that you do not win.

When you roll and even number, the probability of winning is 3/6 or 1/2.
When you want to draw a heart:
The sample space is 12. That is 3 faces and 4 suits. There are 4 hearts in this deck. The probability you get a heart is 4/12 or 1/3. 
When you want to toss a tail: 
Sample space is 2, head and tail. You do not win if you get a heads. The probability is 1/2. The probability of winning is 1/2 * 1/3 * 1/2 = 1/12 Then the probability of losing is 1-1/12 = 11/12