1. ## Conditional Probability

What is the chance that the seventh card from the top of the deck will be a heart?

This question really stumped me. I thought [52_C_6 multiply 13_C_1] and then divided by 52_C_7 would be correct but the probability is greater than 1 which is obviously wrong.

Thank you!

2. Hello, Felipe!

What is the chance that the 7th card from the top of the deck will be a heart?

The deck can be arranged in $\displaystyle 52!$ orderings.

If the 7th card is a Heart, there are 13 choices.
The other 51 cards can be arranged in $\displaystyle 51!$ ways.
. . Hence, there are: $\displaystyle 13\cdot51!$ orderings with the 7th card a Heart.

Therefore: .$\displaystyle P(\text{7th card is a }\heartsuit) \:=\:\dfrac{13\cdot5!!}{52!} \;=\;\dfrac{1}{4}$

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With a little Thought, the answer is obvious.

Consider any card in the deck,
. . the top card, the bottom card, the 37th card . . .

What is the probabiity that it is a Heart?

Since it could any one of the four suits (all equally likely),

. . $\displaystyle P(\text{a card is a }\heartsuit) \:=\:\dfrac{1}{4}$