Probability of drawing the Jack of Hearts?

geocalc33 05/22/2018. 6 answers, 4.954 views

You have a standard deck of cards and randomly take one card away without looking at it and set it aside. What is the probability that you draw the Jack of Hearts from the pile now containing 51 cards? I'm confused by this question because if the card you removed from the pile was the Jack of Hearts then the probability would be zero so I'm not sure how to calculate it.

fleablood 05/22/2018.

The Hard Way

The probability that the first card is not the Jack of Hearts is $\frac {51}{52}$ so the probability that the first card is not the Jack of Hearts and the second card is the Jack of Hearts is $\frac {51}{52}\times \frac 1{51}$.
The probability that the first card is the Jack of Hearts is $\frac 1{52}$ so the probability that the first card is the Jack of Hearts and the second card is the Jack of Hearts is $\frac 1{52}\times 0$.

So the total probability that the second card is Jack of Hearts is:
The probability that the second card is after the first card is not +
The probability that the second card is after the first card already was $$= \frac {51}{52}\times \frac 1{51} + \frac 1{52}\times 0 = \frac 1{52} + 0 = \frac 1{52}$$

That was the hard way.

The Easy Way

The probability that any specific card is any specific value is $\frac 1{52}$. It doesn't matter if it is the first card, the last card, or the 13th card. So the probability that the second card is the Jack of Hearts is $\frac 1{52}$. Picking the first card and not looking at, just going directly to the second card, putting the second card in an envelope and setting the rest of the cards on fire, won't make any difference; all that matters is the second card has a one in $52$ chance of being the Jack of Hearts.

Any thing else just wouldn't make any sense.

The thing is throwing in red herrings like "what about the first card?" doesn't change things and if you actually do try to take everything into account, the result, albeit complicated, will come out to be the same.

Graham Kemp 05/22/2018.

I'm confused by this question because if the card you removed from the pile was the Jack of Hearts then the probability would be zero so I'm not sure how to calculate it.

Well, to go the long way, you need to use the Law of Total Probability.   Letting $X$ be the card you took away, and $Y$ the card you subsequently draw from the remaining deck.

$$\mathsf P(Y=\mathrm J\heartsuit) = \mathsf P(Y=\mathrm J\heartsuit\mid X\neq \mathrm J\heartsuit)~\mathsf P(X\neq \mathrm J\heartsuit)+0\cdot\mathsf P(X=\mathrm J\heartsuit)$$

Well, now, $\mathsf P(X\neq\mathrm J\heartsuit) = 51/52$ is the probability that the card taken is one from the 51 not-jack-of-hearts.

Also $\mathsf P(Y=\mathrm J\heartsuit\mid X\neq \mathrm J\heartsuit)$ is the probability that the subsequent selection is the Jack of Hearts when given that that card is among the 51 remaining cards.

Alternatively, the short way is to consider : When given a shuffled deck of 52 standard cards, what is the probability that the second from the top is the Jack of Hearts?

NicNic8 05/22/2018.

The probability is 1/52.

It doesn't matter that one card was moved somewhere else.

Philip C 05/22/2018.

I think the reason you're getting a bit confused is that you're conflating two different probabilities:

1. Your level of belief that you will choose the Jack of Hearts.
2. Whether or not you will choose the Jack of Hearts.

The first case depends entirely on your knowledge of the state of the deck, and won't be altered until you learn something new about the deck. If you take a random card from the deck and don't look at it, you've learned nothing and your level of belief remains at $1/52$.

The second case depends entirely on the actual, true state of your deck and can be altered if the deck is altered: if the random card you discard is the Jack of Hearts, the true chance of you subsequently randomly selecting that card is zero, otherwise it is $1/51$ as you said.

Incidentally, the second case is analagous to removing a random card and looking at it.

Paul Evans 05/23/2018.

Through extension: You randomly remove 51 cards from a shuffled deck and put them in an envelope. What is the probability the card left behind is the Jack of Hearts?

CiaPan 05/24/2018.

Suppose you don't remove the first card from the pile, but just remember its position, and choose the second card from any of remaining $51$ positions.
This is equivalent to the initial problem, isn't it?

Now, that in turn is equivalent to randomly chosing two positions in a pile and taking one card from the second chosen position.

But in all possible ordered pairs of distinct numbers from the set $\{1,2,\dots 52\}$, each number has exactly the same probability of appearing as the second one in a pair.
This implies you can drop choosing the first position at all — just choose randomly the 'second' position and see if it is the Jack of Hearts there.

This, however, is a simple random selection of a single element from the set, consequently the probability of taking the specific one is $1$ over the number of elements: $\frac 1{52}$.