Probability of Card Draw

[IceXaos]

With a regular deck of 52 cards, how would you find the probability of getting the Ace of Spaces if you took turns until someone got it? This is without reshuffling.

If it was a 1/52 chance each time, I would be able to do it like
(1/52) * (inf. sum ((51/52)^2)^n)

But this would have a limit of 52 trials, and I never learned how to do that. Also, the probability of getting it would change for every card drawn. I also know that it would be different depending on who draws the first card.

Could I get any help?
Thanks

 

[Arkuski]

This is an interesting problem. It turns out that all probabilities are what you expect. For example, if you and a friend took turns finding the ace of spades, both of you have exactly a 50% chance of finding it first. Let's look at this in a little more detail:

First let's say there are 52 people in a line waiting to draw the ace of spades. They each have a 1/52 chance of doing so. The first person has a probablility of (1/52) of drawing the ace of spades. The second person now has a (1/51) chance of drawing an ace of spades from the appended deck, but he also has a (1/52) chance of never seeing the deck if the first person draws the ace of spades. So his chances of drawing the ace of spades are (51/52)(1/51)=(1/52). You can carry out this progression to show similar results for the rest of the line.

Now let's say it's you and a friend taking turns. If every trial has a (1/52) chance of being successful, the probability is evenly split and it is a fair game between you and your friend.

The only way that this game is not fair is if you have n people playing and 52 (mod n) ≠ 0. This logically makes sense because some players would have more possible turns than others.

 

[tiny-tim]

Hi IceXaos!

With a regular deck of 52 cards, how would you find the probability of getting the Ace of Spaces if you took turns until someone got it? This is without reshuffling.

Simplest way:

Number the cards 1 to 52.

You choose 26, your opponent chooses 26.

It doesn't matter which order you choose them (you could choose all your 26 cards first, for example) …

if the Ace of Spades is in your half, you win, if it isn't, you don't! ​

 

[IceXaos]

I didn't even attempt the simple way of looking at it. That's kind'a unexpected.

Thanks guys.

 

[blue_raver22]

for your question! I would approach this problem by using the principles of probability and statistics. In this case, we are dealing with a situation where the probability of getting the Ace of Spaces changes with each card drawn, as the deck is not being reshuffled. This means that the probability of getting the Ace of Spaces on the first draw is 1/52, but as more cards are drawn, the probability changes.

To find the overall probability of getting the Ace of Spaces, we can use the concept of conditional probability. This means finding the probability of getting the Ace of Spaces given that a certain number of cards have already been drawn. For example, the probability of getting the Ace of Spaces on the second draw would be 1/51, as there are now 51 cards remaining in the deck.

To find the overall probability, we would need to consider all possible scenarios where the Ace of Spaces could be drawn. This would involve calculating the conditional probabilities for each possible number of cards drawn, from 1 to 52. We can then use the principle of addition to find the total probability by adding up all of these individual probabilities.

In this case, it would be too complicated to calculate the exact probability using this method, as there are many different scenarios to consider. However, we can use statistical techniques such as simulation or sampling to estimate the probability. This involves conducting multiple trials, where we take turns drawing cards until someone gets the Ace of Spaces, and then recording the number of cards it took. By repeating this process many times, we can estimate the overall probability.

Overall, the probability of getting the Ace of Spaces in a game where the deck is not reshuffled after each turn would be less than 1/52, as the probability changes with each card drawn. However, by using statistical methods, we can estimate this probability and get a better understanding of the likelihood of getting the Ace of Spaces in this scenario.