1. **Problem Statement:** James rolls a fair six-sided die and draws one card from 17 cards numbered 1 to 17. We are asked to solve four parts related to this experiment. 2. **(a) Number of possible outcomes:** - The die has 6 outcomes. - The card deck has 17 outcomes. - Since these are independent events, total outcomes = $6 \times 17 = 102$. 3. **(b) Probability that the die roll and card draw are the same number:** - Possible matching numbers are 1 through 6 (since die max is 6). - For each matched number, one outcome. - So, favorable outcomes = 6. - Probability = $\frac{6}{102} = \frac{1}{17}$. 4. **(c) Probability both results are square numbers:** - Square numbers on die: 1, 4 (since $1^2=1$, $2^2=4$; $3^2=9$ not on die). - Square numbers on cards between 1 and 17: $1,4,9,16$. - Possible matching pairs: (1,1), (4,4). - Total favorable outcomes = $2$. - Probability = $\frac{2}{102}= \frac{1}{51}$. 5. **(d) Probability James loses (result is 4 or less on either die or card):** - Die outcomes $\leq 4$ = 4 outcomes. - Card outcomes $\leq 4$ = 4 outcomes. - Total outcomes = 102. - Number of outcomes where die $\leq 4 = 4 \times 17 = 68$. - Number where card $\leq 4 = 6 \times 4 = 24$. - Double counted outcomes where both die and card $\leq 4 = 4 \times 4 = 16$. - Use inclusion-exclusion: Favorable outcomes = $68 + 24 - 16 = 76$. - Probability = $\frac{76}{102} = \frac{38}{51}$. **Final answers:** (a) 102 (b) $\frac{1}{17}$ (c) $\frac{1}{51}$ (d) $\frac{38}{51}$