Suppose that you toss a coin and roll a die. The sample space is shown below. (Enter the probabilities as fractions.) (a) What is the probability of obtaining tails and an even number? $1 / 2$ s (b) What is the probability of obtaining tails or an even number? $1 / 4$

Step 1 ：The sample space for this problem is the set of all possible outcomes. Since we are tossing a coin and rolling a die, there are 2 possible outcomes for the coin (heads or tails) and 6 possible outcomes for the die (1, 2, 3, 4, 5, 6). Therefore, the total number of possible outcomes is 2 * 6 = 12.

Step 2 ：To find the probability of obtaining tails and an even number, we need to count the number of outcomes that satisfy both conditions. There are 3 even numbers (2, 4, 6) and 1 outcome for tails, so there are 3 * 1 = 3 outcomes that satisfy both conditions. Therefore, the probability is 3 / 12 = 1 / 4.

Step 3 ：To find the probability of obtaining tails or an even number, we need to count the number of outcomes that satisfy either condition. There are 3 even numbers (2, 4, 6) and 2 outcomes for tails (since the coin can land on tails whether the die shows an even or odd number), so there are 3 + 2 = 5 outcomes that satisfy either condition. Therefore, the probability is 5 / 12.

Step 4 ：The probability of obtaining tails and an even number is \(\boxed{\frac{1}{4}}\).

Step 5 ：The probability of obtaining tails or an even number is \(\boxed{\frac{5}{12}}\).