Iwo cards are drawn at random with replacement. Find each probability. 7. \( P( \) M...then \( A) 1 / 24 \) 9. \( P(1 \). then \( T) 1 / 7.2 \) 8. \( P( \) not \( M \), then \( A) 5 / 2 U \) 10. \( P \) (vowel, then vowel) Iwo cards are drawn at random without replacement. Find each probability. 11. \( P(M \), then \( A) \) 13. \( P(M \). then \( M) \) 12. \( P(A \), then \( A) \) 15. \( P \) (vowel. then vowel) 14. \( P(E \), then \( T) \) 16. \( P( \) vowel, then \( M) \)

To solve the probability problems, we need to clarify the context and the total number of cards involved. Let's assume we have a standard deck of 26 cards (A-Z) for the calculations. ### With Replacement 1. **Problem 7: \( P(M \text{ then } A) \)** - The probability of drawing \( M \) first is \( \frac{1}{26} \). - The probability of drawing \( A \) second (with replacement) is also \( \frac{1}{26} \). - Therefore, \( P(M \text{ then } A) = P(M) \times P(A) = \frac{1}{26} \times \frac{1}{26} = \frac{1}{676} \). 2. **Problem 9: \( P(1 \text{ then } T) \)** - Assuming \( 1 \) refers to a specific card (not in a standard deck), we will treat it as a unique event. - The probability of drawing \( 1 \) is \( \frac{1}{26} \) (if it exists). - The probability of drawing \( T \) is \( \frac{1}{26} \). - Therefore, \( P(1 \text{ then } T) = \frac{1}{26} \times \frac{1}{26} = \frac{1}{676} \). 3. **Problem 8: \( P(\text{not } M \text{ then } A) \)** - The probability of not drawing \( M \) is \( \frac{25}{26} \). - The probability of drawing \( A \) is \( \frac{1}{26} \). - Therefore, \( P(\text{not } M \text{ then } A) = \frac{25}{26} \times \frac{1}{26} = \frac{25}{676} \). 4. **Problem 10: \( P(\text{vowel then vowel}) \)** - There are 5 vowels (A, E, I, O, U). - The probability of drawing a vowel first is \( \frac{5}{26} \). - The probability of drawing a vowel second (with replacement) is also \( \frac{5}{26} \). - Therefore, \( P(\text{vowel then vowel}) = \frac{5}{26} \times \frac{5}{26} = \frac{25}{676} \). ### Without Replacement 1. **Problem 11: \( P(M \text{ then } A) \)** - The probability of drawing \( M \) first is \( \frac{1}{26} \). - After drawing \( M \), there are 25 cards left, and the probability of drawing \( A \) is \( \frac{1}{25} \). - Therefore, \( P(M \text{ then } A) = \frac{1}{26} \times \frac{1}{25} = \frac{1}{650} \). 2. **Problem 13: \( P(M \text{ then } M) \)** - The probability of drawing \( M \) first is \( \frac{1}{26} \). - After drawing \( M \), there are no more \( M \) cards left, so the probability of drawing \( M \) again is \( 0 \). - Therefore, \( P(M \text{ then } M) = 0 \). 3. **Problem 12: \( P(A \text{ then } A) \)** - The probability of drawing \( A \) first is \( \frac{1}{26} \). - After drawing \( A \), there are 25 cards left, and the probability of drawing \( A \) again is \( 0 \). - Therefore, \( P(A \text{ then } A) = 0 \). 4. **Problem 15: \( P(\text{vowel then vowel}) \)** - The probability of drawing a vowel first is \( \frac{5}{26} \). - After drawing a vowel, there are 4 vowels left out of 25 cards. - Therefore, \( P(\text{vowel then vowel}) = \frac{5}{26} \times \frac{4}{25} = \frac{20}{650} = \frac{2}{65} \). 5. **Problem 14: \( P(E \text{ then } T) \)** - The probability of drawing \( E \) first is \( \frac{1}{26} \). - After drawing \( E \), the probability of drawing \( T \) is \( \frac{1}{25} \). - Therefore, \( P(E \text{ then } T) = \frac{1}{26} \times \frac{1}{25} = \frac{1}{650} \). 6. **Problem 16: \( P(\text{vowel then } M) \)** - The probability of drawing a vowel first is \( \frac{5}{26} \). - After drawing a vowel, the probability of drawing \( M \) is \( \frac{1}{25} \). - Therefore, \( P(\text{vowel then } M) = \frac{5}{26} \times \frac{1}{25} = \frac{5}{650} = \frac{1}{130} \). ### Summary of Probabilities - With Replacement: - \( P(M \text{ then } A) = \frac{1}{676} \) - \( P(1 \text{ then } T) = \frac{1}{676} \) - \( P(\text{not } M \text{ then } A) = \frac{25}{676} \) - \( P(\text{vowel then vowel}) = \frac{25}{676} \) - Without Replacement: - \( P(M \text{ then } A) = \frac{1}{650} \) - \( P(M \text{ then } M) = 0 \) - \( P(A \text{ then } A) = 0 \) - \( P(\text{vowel then vowel}) = \frac{2}{65} \) - \( P(E \text{ then } T) = \frac{1}{650} \) - \( P(\text{vowel then } M) = \frac{1}{130} \)