115 A game requires you to toss a 10 -sided numbered solid and a 6 -sided numbered solid to determine how to move on a game board. Find the following probabilities. a. \( P \) (same number on both) b. \( P \) (odd, even) or \( P \) (even, odd)

To get the same number on both dice, we can only have the numbers 1 through 6 match, since the 10-sided die has numbers up to 10. There are 6 favorable outcomes (1 through 6) and a total of 60 possible outcomes (10 x 6). So, \( P(same number) = \frac{6}{60} = \frac{1}{10} \). For the second probability, we determine all combinations of odd and even results. There are 5 odd and 5 even outcomes on the 10-sided die. When considering the overall possibilities, \( P(odd, even) \) plus \( P(even, odd) \), we see that we have 5 choices on the 10-sided and 3 choices on the 6-sided die (3 odd + 3 even) yielding a total of \( 5 \cdot 3 + 3 \cdot 5 = 30 \) favorable outcomes out of 60. Thus, \( P(odd, even) \text{ or } P(even, odd) = \frac{30}{60} = \frac{1}{2} \).