Let's solve the two parts of the problem step-by-step.

(a) Finding $P(\text{FLUSH})$

A flush is a hand where all five cards have the same suit. In a standard deck of 52 cards, there are 4 suits (hearts, diamonds, clubs, and spades), each containing 13 cards.

1. Number of ways to choose a flush:
   To form a flush, we first choose the suit (4 choices), and then we choose 5 cards from the 13 cards available in that suit.

   $\text{Number of ways to choose 5 cards from 13 of the same suit} = \binom{13}{5}$

2. Total number of possible hands:
   The total number of ways to choose any 5 cards from the 52-card deck is:

   $\binom{52}{5}$

3. Calculating the probability:
   The probability of drawing a flush is the ratio of the number of ways to draw a flush to the total number of possible hands.

   = \frac{4 \cdot \binom{13}{5}}{\binom{52}{5}}$$

Now, let's compute this.

$\binom{13}{5} = \frac{13!}{5!(13-5)!} = \frac{13 \times 12 \times 11 \times 10 \times 9}{5 \times 4 \times 3 \times 2 \times 1} = 1287$

$\binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52 \times 51 \times 50 \times 49 \times 48}{5 \times 4 \times 3 \times 2 \times 1} = 2598960$

Substitute these values:

$P(\text{FLUSH}) = \frac{4 \times 1287}{2598960} = \frac{5148}{2598960}$

$P(\text{FLUSH}) \approx 0.00198$

So, the probability of getting a flush is approximately 0.198%.

(b) Finding $P(\text{SPECIAL})$

The SPECIAL event is defined as a hand containing any five cards that are among the ranks Ace (A), King (K), Queen (Q), and Jack (J), regardless of suits.

1. Number of ways to form a SPECIAL hand:

   • We first identify the cards involved: A, K, Q, J. There are 4 ranks and 4 suits, meaning there are a total of $4 \times 4 = 16$ cards to choose from (4 Aces, 4 Kings, 4 Queens, 4 Jacks).
   • We need to select 5 cards from these 16.

   $\text{Number of SPECIAL hands} = \binom{16}{5}$

2. Total number of possible hands:
   As calculated previously:

   $\binom{52}{5} = 2598960$

3. Calculating the probability:
   The probability of drawing a SPECIAL hand is:

   $P(\text{SPECIAL}) = \frac{\binom{16}{5}}{\binom{52}{5}}$

Now, compute $\binom{16}{5}$:

$\binom{16}{5} = \frac{16!}{5!(16-5)!} = \frac{16 \times 15 \times 14 \times 13 \times 12}{5 \times 4 \times 3 \times 2 \times 1} = 4368$

Substitute these values:

$P(\text{SPECIAL}) = \frac{4368}{2598960}$

$P(\text{SPECIAL}) \approx 0.00168$

So, the probability of getting a SPECIAL hand is approximately 0.168%.

Summary of Results

• (a) $P(\text{FLUSH})$: 0.198%
• (b) $P(\text{SPECIAL})$: 0.168%

Would you like more details or have any questions?

Here are 5 related questions that could expand your understanding:

1. What is the probability of getting a straight in a 5-card hand?
2. How would the probability of getting a flush change if drawing 7 cards instead of 5?
3. What is the probability of getting a full house in a 5-card hand?
4. How does the calculation change if the deck has jokers included?
5. What is the probability of getting four of a kind in a 5-card hand?

Tip: When solving probability problems involving cards, always consider the total possible outcomes and how many of those outcomes satisfy the event condition.