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 $$ M) $$

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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: $$ P(M 	ext{ then } A) $$**
   - The probability of drawing $$ M $$ first is $$ \frac{1}{26} $$.
   - The probability of drawing $$ A $$ second (with replacement) is also $$ \frac{1}{26} $$.
   - Therefore, $$ P(M 	ext{ then } A) = P(M) 	imes P(A) = \frac{1}{26} 	imes \frac{1}{26} = \frac{1}{676} $$.

2. **Problem 9: $$ P(1 	ext{ then } T) $$**
   - 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, $$ P(1 	ext{ then } T) = \frac{1}{26} 	imes \frac{1}{26} = \frac{1}{676} $$.

3. **Problem 8: $$ P(	ext{not } M 	ext{ then } A) $$**
   - The probability of not drawing $$ M $$ is $$ \frac{25}{26} $$.
   - The probability of drawing $$ A $$ is $$ \frac{1}{26} $$.
   - Therefore, $$ P(	ext{not } M 	ext{ then } A) = \frac{25}{26} 	imes \frac{1}{26} = \frac{25}{676} $$.

4. **Problem 10: $$ P(	ext{vowel then vowel}) $$**
   - 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, $$ P(	ext{vowel then vowel}) = \frac{5}{26} 	imes \frac{5}{26} = \frac{25}{676} $$.

### Without Replacement

1. **Problem 11: $$ P(M 	ext{ then } A) $$**
   - 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, $$ P(M 	ext{ then } A) = \frac{1}{26} 	imes \frac{1}{25} = \frac{1}{650} $$.

2. **Problem 13: $$ P(M 	ext{ then } M) $$**
   - 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, $$ P(M 	ext{ then } M) = 0 $$.

3. **Problem 12: $$ P(A 	ext{ then } A) $$**
   - 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, $$ P(A 	ext{ then } A) = 0 $$.

4. **Problem 15: $$ P(	ext{vowel then vowel}) $$**
   - 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, $$ P(	ext{vowel then vowel}) = \frac{5}{26} 	imes \frac{4}{25} = \frac{20}{650} = \frac{2}{65} $$.

5. **Problem 14: $$ P(E 	ext{ then } T) $$**
   - The probability of drawing $$ E $$ first is $$ \frac{1}{26} $$.
   - After drawing $$ E $$, the probability of drawing $$ T $$ is $$ \frac{1}{25} $$.
   - Therefore, $$ P(E 	ext{ then } T) = \frac{1}{26} 	imes \frac{1}{25} = \frac{1}{650} $$.

6. **Problem 16: $$ P(	ext{vowel then } M) $$**
   - 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, $$ P(	ext{vowel then } M) = \frac{5}{26} 	imes \frac{1}{25} = \frac{5}{650} = \frac{1}{130} $$.

### Summary of Probabilities
- With Replacement:
  - $$ P(M 	ext{ then } A) = \frac{1}{676} $$
  - $$ P(1 	ext{ then } T) = \frac{1}{676} $$
  - $$ P(	ext{not } M 	ext{ then } A) = \frac{25}{676} $$
  - $$ P(	ext{vowel then vowel}) = \frac{25}{676} $$

- Without Replacement:
  - $$ P(M 	ext{ then } A) = \frac{1}{650} $$
  - $$ P(M 	ext{ then } M) = 0 $$
  - $$ P(A 	ext{ then } A) = 0 $$
  - $$ P(	ext{vowel then vowel}) = \frac{2}{65} $$
  - $$ P(E 	ext{ then } T) = \frac{1}{650} $$
  - $$ P(	ext{vowel then } M) = \frac{1}{130} $$
Here are the probabilities for drawing two cards: **With Replacement:** - $$ P(M 	ext{ then } A) = \frac{1}{676} $$ - $$ P(1 	ext{ then } T) = \frac{1}{676} $$ - $$ P(	ext{not } M 	ext{ then } A) = \frac{25}{676} $$ - $$ P(	ext{vowel then vowel}) = \frac{25}{676} $$ **Without Replacement:** - $$ P(M 	ext{ then } A) = \frac{1}{650} $$ - $$ P(M 	ext{ then } M) = 0 $$ - $$ P(A 	ext{ then } A) = 0 $$ - $$ P(	ext{vowel then vowel}) = \frac{2}{65} $$ - $$ P(E 	ext{ then } T) = \frac{1}{650} $$ - $$ P(	ext{vowel then } M) = \frac{1}{130} $$

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