# Deck of cards without replacement

• MHB
• jtkerr
In summary, the odds against you being in this situation - having drawn 49 consecutive incorrect cards without replacement are 3/52.
jtkerr
I have a standard 52 card deck of cards. I am trying to draw the ace of spades. I draw cards one at a time. I do not replace each incorrect card. I am about to draw my 50th card having drawn 49 consecutive cards without ever drawing the ace of spades. There are three cards left. I know that I have a 33% chance of drawing the ace of spades now. My question:

What are the odds against me being in this situation - having drawn 49 consecutive incorrect cards without replacement?

jtkerr said:
I have a standard 52 card deck of cards. I am trying to draw the ace of spades. I draw cards one at a time. I do not replace each incorrect card. I am about to draw my 50th card having drawn 49 consecutive cards without ever drawing the ace of spades. There are three cards left. I know that I have a 33% chance of drawing the ace of spades now. My question:

What are the odds against me being in this situation - having drawn 49 consecutive incorrect cards without replacement?

Hi jtkerr,

Welcome to MHB! :)

So you are searching for 1 specific card out of 52 cards without replacement. For the 1st round there are 51 cards that aren't your card out of 52. So the probability of not drawing the As on round 1 is 51/52. Now for round 2 there are 51 cards left. The probability of not drawing the As is now 50/51. You can repeat this process all the way until there are 3 cards left and if you multiply all of that out you should get your answer.

Intuitively this should be very small, as drawing 49 out of 52 cards should get your specific card most of the time. What do you get as an answer when you try this approach?

There are, to begin with, 52 cards, 51 of which are "incorrect" (not the ace of spades). The probability of drawing an "incorrect" card is $$\displaystyle \frac{51}{52}$$. If you do draw an "incorrect" card, there are 51 cards left, 50 of which are "incorrect". The probability of drawing an "incorrect" card now is $$\displaystyle \frac{50}{51}$$. On the 49th such draw there will be 52- 48= 4 cards left, 3 of which are "incorrect". The probability of drawing an "incorrect" card on the 49th draw is $$\displaystyle \frac{3}{4}$$.

The probability of drawing 49 consecutive "incorrect" cards is
$$\displaystyle \frac{51}{52}\frac{50}{51}\frac{49}{50}\cdot\cdot\cdot \frac{4}{5}\frac{3}{4}$$.

You can see that the numerator in each fraction will cancel the denominator in the next fraction leaving $$\displaystyle \frac{3}{52}$$.

Hi guys, can someone explain, how can I find out the probability and payout of hand, and than how to calculate the expected return of each hand? Let's say I've got the pair of Aces. There are 16215 possible combinations. Is it possible to count it somehow without using any computer or software?
In case of royal flush, there are only 47 possible outcomes. 27 times we won’t hit anything. 8 times we will hit Jacks or Better, 3 times a straight, 8 times a flush and one time a Royal flush.
I get it till here... How do I get that ,,expected value of the hand?'' If I don't know if someone will raise the bets or not...
Can someone explain it to me?

poker strategy source I've used.

## 1. What is a deck of cards without replacement?

A deck of cards without replacement refers to a set of playing cards where each card is drawn and not replaced back into the deck before the next card is drawn. This means that the probability of drawing a specific card changes as cards are drawn from the deck.

## 2. How many cards are in a deck without replacement?

A deck of cards without replacement typically consists of 52 cards, which are divided into four suits (clubs, diamonds, hearts, and spades) with 13 cards in each suit. However, some variations may include additional cards such as jokers.

## 3. What is the probability of drawing a specific card from a deck without replacement?

The probability of drawing a specific card from a deck without replacement depends on the number of cards remaining in the deck and the number of cards of the same suit or value as the specific card. For example, if there are 26 cards remaining in the deck and 4 cards of the same suit as the specific card, the probability would be 4/26 or approximately 15%.

## 4. How does drawing without replacement affect the probability of subsequent draws?

As cards are drawn without replacement, the probability of drawing a specific card changes. This is because the number of cards remaining in the deck changes and the probability is affected by the cards that have been previously drawn. For example, if a card of a specific suit is drawn, the probability of drawing another card of the same suit decreases as there are fewer cards of that suit remaining in the deck.

## 5. What are some examples of games that use a deck of cards without replacement?

Some popular games that use a deck of cards without replacement include poker, blackjack, and bridge. These games rely on the changing probabilities of drawing specific cards from the deck to create challenging and strategic gameplay.

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