$\textbf{The Problem:}$ Suppose I ask you to pick any four cards at random from a deck of $52,$ without replacement, and bet you one dollar that at least one of the four is a face card (i.e., Jack, Queen, or King). Should you take the bet?
$\textbf{My Thoughts:}$ I compute the probability of not drawing a face card in four draws without replacement as follows
$$\mathsf{P}(\text{no face card})=\frac{40}{52}\cdot\frac{40}{51}\cdot\frac{40}{50}\cdot\frac{40}{49}=0.39.$$
Therefore we have
$$\mathsf{P}(\text{at least one face card})=1-\mathsf{P}(\text{no face card})=0.61.$$
Which implies that I should take the bet, since the probability of winning is greater than $50$%.
Could anyone provide feedback on my solution to the problem?
Thank you for your time, and appreciate any help or feedback.
Best Answer
You are a little off. The probability of drawing four cards, none of which is a face card, is $$\frac{40 \times 39 \times 38 \times 37}{52 \times 51 \times 50 \times 49}$$
Another way to look at the problem, which leads to the same answer, is that there are $\binom{52}{4}$ four-card hands, all of which we assume are equally likely. Of these, $\binom{40}{4}$ consist of non-face cards. So the probability of drawing a four-card hand of non-face cards is $${\binom{40}{4}}/{\binom{52}{4}}$$