11. (40 Points) You pick 5 cards at random from a deck and have the 9, 10, J, Q of Diamonds and the Ace of Spades. You throw away the Ace of Spades and draw one card from the deck (no A of Spades is possible). What are the odds that you make either a straight 8910J, 910JQK, a flush (all diamonds) or a straight flush 8910JQ or 9110JQK all Diamonds. Do not triple count, a straight, straight flush and a flush, all count as a win. Show your work.

To solve this problem, we need to calculate the probability of drawing a card that would complete either a straight, a flush, or a straight flush given the current hand of \(\{9, 10, J, Q\}\) of Diamonds and the Ace of Spades being discarded. We then analyze the potential outcomes for the new card and determine the odds of achieving any of the desired combinations.

### Step 1: Understand the Initial Hand and Desired Outcomes

Your initial hand after discarding the Ace of Spades is:
\[
\{9\text{♦}, 10\text{♦}, J\text{♦}, Q\text{♦}\}
\]
You are missing one card to form:
1. **A straight**: Either \(8\) of any suit or \(K\) of any suit.
2. **A flush**: Any Diamond other than \(A\) (since it was discarded).
3. **A straight flush**: Either \(8\text{♦}\) or \(K\text{♦}\).

### Step 2: Determine the Number of Possible Outcomes

Since the Ace of Spades has been removed, 51 cards remain in the deck.

### Step 3: Calculate the Number of Winning Cards

1. **Straight**:
   - \(8\) (any suit) and \(K\) (any suit): There are 4 suits for \(8\) and 4 suits for \(K\), resulting in \(4 + 4 = 8\) cards that will give a straight.

2. **Flush**:
   - Any Diamond except the Ace, since it's already discarded. There are 13 Diamonds in total, and you already have 4 of them, and the Ace is removed. Thus, 8 Diamonds remain in the deck.

3. **Straight Flush**:
   - Either \(8\text{♦}\) or \(K\text{♦}\) completes the straight flush. So, there are 2 specific cards (the \(8\text{♦}\) and \(K\text{♦}\)).

### Step 4: Consider Overlapping Outcomes

- Cards that give a straight flush (8♦ and K♦) also count towards making a straight and a flush. 
- Therefore, we must not double or triple count these outcomes.

**Total Winning Cards**:
- Total number of cards that complete a straight or flush: \(8 + 8 = 16\).
- Subtract the overlap (the \(8\text{♦}\) and \(K\text{♦}\)) twice since they are counted in both straight and flush. This gives:
  \[
  \text{Winning Cards} = 16 - 2 = 14
  \]

### Step 5: Calculate the Probability

The probability of drawing a winning card is:
\[
P(\text{win}) = \frac{14}{51}
\]

### Step 6: Calculate the Odds

The odds in favor of making either a straight, flush, or straight flush are:
\[
\text{Odds in favor} = \frac{\text{Number of Winning Outcomes}}{\text{Number of Losing Outcomes}} = \frac{14}{51 - 14} = \frac{14}{37}
\]

### Final Answer:
The odds that you make either a straight, flush, or straight flush by drawing one more card are \( \frac{14}{37} \).

---

Would you like to go into any specific detail or do you have any questions? Here are some related questions that might interest you:

1. How do you calculate the probability of drawing a specific card from a deck?
2. What are the odds of drawing a card of a specific suit after one card is removed?
3. How do odds differ from probability in card games?
4. Can you explain how to avoid double counting in probability problems?
5. How do straight flush odds compare to general flush odds?
6. What is the probability of completing a royal flush from four suited cards?
7. How does discarding one card change the probability of drawing another specific card?
8. How are combinations and permutations used in calculating card game probabilities?

**Tip**: In probability problems, always check for overlapping conditions to avoid double-counting. This is crucial in problems involving combinations like straights and flushes.

Understand the probability of completing a straight, flush, or straight flush in a card game scenario involving drawing from a deck after discarding specific cards. Learn the step-by-step solution and formula application.

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