# Probability Question from my textbook

A pack contains m cards, labeled 1, 2,....,m. The cards are dealt out in a random order, one by one. Given that the label of the kth card dealt is the largest of the first k cards, what is the probability that it is also the largest in the whole pack?

Reading probability myself and stuck at some questions.

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(You might want this to get double-checked by someone else; my probability is a bit rusty.)

Recall the formula for conditional probability: \begin{align} P(A \cap B) &= P(A|B)P(B) \\ P(A|B) &= \frac{P(A \cap B)}{P(B)} \end{align}

Here, let $A$ be the event that the $k$th card is the largest of the $m$ cards, and $B$ be the event that the $k$th card is the largest of the $k$ cards drawn.

Since $A$ implies $B$, $P(A \cap B) = P(A)$. So the probability is $1/m$. The probability of event $B$, that the $k$th card is the largest of $k$ cards, without knowing any other information, is $1/k$.

Then $$P(A|B) = \frac{1/m}{1/k} = \frac{k}{m}.$$

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The probability is equal to the probability that the largest of m cards is one of the first k cards, which is $\frac{k}{m}$.

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