Anne, Joe and Lynn, in this order, will take turns to draw two cards from a standard deck of 52 cards. Note that each draw of 2 cards is w/o replacement but draws of different turns are with replacement (the preceding two cards will be put back and the deck will be reshuffled.) If they will take turns until four suit matches (two cards of the same suit) are observed what is the probability that each of them will take five turns.

To determine the probability that each of them will take five turns to observe four suit matches, we can break down the problem into steps:

Step 1: Calculate the probability of the first player (Anne) taking five turns to observe four suit matches.

In order for Anne to take five turns and observe four suit matches, the following conditions must be met:

- The 1st turn drawing 2 cards: The probability of drawing two cards of the same suit is 1 since it is the first turn.
- The 2nd turn drawing 2 cards: The probability of drawing two cards of the same suit is 3/51 (3 cards of the same suit remaining in the deck, out of 51 cards remaining in total).
- The 3rd turn drawing 2 cards: The probability of drawing two cards of the same suit is 7/49 (7 cards of the same suit remaining in the deck, out of 49 cards remaining in total).
- The 4th turn drawing 2 cards: The probability of drawing two cards of the same suit is 11/47 (11 cards of the same suit remaining in the deck, out of 47 cards remaining in total).
- The 5th turn drawing 2 cards: The probability of drawing two cards of the same suit is 15/45 (15 cards of the same suit remaining in the deck, out of 45 cards remaining in total).

To find the overall probability of Anne taking five turns to observe four suit matches, we multiply these individual probabilities together:

1 * (3/51) * (7/49) * (11/47) * (15/45) = 231/280280

Step 2: Calculate the probability of the second player (Joe) taking five turns to observe four suit matches.

Like in the case of Anne, Joe has the same probability of drawing two cards of the same suit in each turn:

1 * (3/51) * (7/49) * (11/47) * (15/45) = 231/280280

Step 3: Calculate the probability of the third player (Lynn) taking five turns to observe four suit matches.

Following the same logic:

1 * (3/51) * (7/49) * (11/47) * (15/45) = 231/280280

Step 4: Calculate the overall probability.

To find the overall probability, we multiply the probabilities of each player taking five turns together:

(231/280280) * (231/280280) * (231/280280) = 12146751/2263642501248000

Therefore, the probability that each of them will take five turns to observe four suit matches is 12146751/2263642501248000, which is approximately 0.005367.

To find the probability that each of them will take exactly five turns, we need to determine the total number of possible outcomes and the number of favorable outcomes.

Let's break down the problem step by step:

Step 1: Determine the number of ways each person can draw two cards from the deck.
For the first draw, each person can choose any two cards from the 52-card deck, so there are (52 choose 2) = 1326 possible ways.
For the second draw, each person can again choose any two cards, but this time they should consider the cards drawn by the previous players. Since the previously drawn cards are replaced back into the deck and shuffled, the number of possible ways is still (52 choose 2) = 1326.
Similarly, for the third, fourth, and fifth draws, the number of possible ways remains the same - (52 choose 2) = 1326.

Step 2: Determine the number of favorable outcomes.
To have exactly four suit matches at the end of five turns, each person should draw two cards of the same suit in the last round.
In the first four turns, the players can draw any cards, so there are no restrictions.
But in the fifth and final turn, the first player needs to draw two cards of the same suit. The probability of this happening is (13 choose 1)*(4 choose 2) / (52 choose 2), where (13 choose 1) represents choosing one suit out of the four available, and (4 choose 2) represents choosing two cards from that suit.

Similarly, for the second player, the probability of drawing two cards of the same suit in the final round is also [(13 choose 1)*(4 choose 2)] / (52 choose 2).
The same applies to the third player and the fourth player.

Step 3: Calculate the probability.
Since the outcomes of each player's turns are independent, we need to multiply the probabilities together.
The probability that each player takes exactly five turns is:
[(13 choose 1)*(4 choose 2) / (52 choose 2)]^4

Note: The above expression is valid if you consider that each player has to take exactly five turns. If you want to allow some players to take more or fewer turns, the calculation of probabilities will be different.

Evaluate the expression to find the exact value of the probability.