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Suppose you draw $5$ cards out of a deck of $52$ and get $2$ spades and $3$ hearts. What is the probability that the first card drawn was a spade?


Is this question easier than it looks? Since we're only interested in the first card is it not simply $\frac{13}{52}$?

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    $\begingroup$ Do not forget that we have the additional phrase in the question given that we got 2 spades and 3 hearts. Instead of thinking about drawing a card from a 52card deck, think about drawing it from a five card deck with two spades and 3 hearts comprising the entirety of the deck. This is a question on conditional probability. $\endgroup$ – JMoravitz Sep 24 '17 at 23:19
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    $\begingroup$ Oh! So the answer is $\frac{2}{5}$? $\endgroup$ – Hello Sep 24 '17 at 23:20
  • $\begingroup$ @JMoravitz One quick thing. Are you sure that's how it is? Isn't it asking what the probability that the first card of the 5 drawn cards was a spade? That's why I was thinking $\frac{13}{52}$. Wouldn't your approach give an answer of picking a spade from the actual 5 cards? $\endgroup$ – Hello Sep 24 '17 at 23:40
  • $\begingroup$ To aid with intuition, suppose it asked the probability that the first card chosen was a diamond. that is clearly $0$, yes? $\endgroup$ – lulu Sep 24 '17 at 23:43
  • $\begingroup$ @lulu The first card from what exactly? From the original 52? Or from the 5? $\endgroup$ – Hello Sep 24 '17 at 23:44
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This problem actually is easier than it looks. Probability is all about making the best prediction using all the information you have, so in this case, you should only focus on the five cards whose values you already know.

The probability that the first card drawn was a spade is the number of ways for a spade to come first divided by the total number of ways to draw those five cards in an ordered sequence.


The number of ways to draw a spade first is

$$2\cdot{}_4P_4$$ You put the spade in first position, shuffle the other four around, add up the number of different permutations, and then repeat this for the other spade.

The number of ways to draw the five cards in an ordered sequence is

$${}_5P_5$$ This is just your average permutation.

The final answer is

$$\frac25$$

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