1. **Problem statement:** We have 5 identical cards numbered 2, 3, 4, 5, 6. Two cards are drawn one after the other with replacement, forming a two-digit number. We want to find the probability that the units digit is prime or the tens digit is odd. 2. **Total possible outcomes:** Since the draws are with replacement, each draw has 5 possibilities. The total number of two-digit numbers formed is $$5 \times 5 = 25$$. 3. **Define events:** - Let $A$ be the event "units digit is prime". - Let $B$ be the event "tens digit is odd". 4. **Identify prime digits:** The prime digits among the cards are 2, 3, 5. So units digit prime means units digit is in $\{2,3,5\}$. 5. **Identify odd digits:** The odd digits among the cards are 3, 5. So tens digit odd means tens digit is in $\{3,5\}$. 6. **Calculate $P(A)$:** Units digit prime means units digit is 2, 3, or 5. - For each tens digit (5 possibilities), units digit can be 3 possibilities. - So, $|A| = 5 \times 3 = 15$. 7. **Calculate $P(B)$:** Tens digit odd means tens digit is 3 or 5. - For each tens digit (2 possibilities), units digit can be any of 5. - So, $|B| = 2 \times 5 = 10$. 8. **Calculate $P(A \cap B)$:** Numbers where tens digit is odd (3 or 5) and units digit is prime (2, 3, 5). - Tens digit: 2 possibilities. - Units digit: 3 possibilities. - So, $|A \cap B| = 2 \times 3 = 6$. 9. **Use the formula for union:** $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ 10. **Calculate probabilities:** $$P(A) = \frac{15}{25} = 0.6$$ $$P(B) = \frac{10}{25} = 0.4$$ $$P(A \cap B) = \frac{6}{25} = 0.24$$ 11. **Final probability:** $$P(A \cup B) = 0.6 + 0.4 - 0.24 = 0.76$$ **Answer:** The probability that the units digit is prime or the tens digit is odd is $$\boxed{0.76}$$.