Question

Sherwin Betlotz, the tricky gambler, will bet even money that you cant pick three cards from a 52-cards deck without getting at least one of the twelve face cards. Would you bet with him?

Answer 1

$Probability=\frac{n\left(Likelyevents\right)}{n\left(Totalevents\right)}$

ie,P(First card$\ne$Face card)$=\frac{52-12}{52}=\frac{10}{13}$

P(Second card$\ne$Face card)$=\frac{51-12}{51}=\frac{39}{51}$ ($\because$ Events are dependent. After we've taken the first card only 51 cards left)

Similarly, P(Third card$\ne$Face card)$=\frac{50-12}{50}=\frac{19}{25}$

$\therefore$Total Probability of not getting a face card on picking three cards from a deck$=\frac{10}{13}\times \frac{39}{51}\times \frac{19}{25}=\frac{2\times 19}{5\times 17}=\frac{38}{85}<\frac{1}{2}$

So, as my chances are below fair(50%), you should not bet with him else you may lose the bet.