You draw three cards into your hand out of 54. The goal number of your hand total is 4, what is your probability of winning in your first draw?

Question

You play a game where all number cards (2-10) have a value equal to their number, ace has a value of 1, face cards have a value of 10 and joker can have the same value as any card of your choice. To play the game, you draw three cards into your hand, and you win if your hand total (sum of values) meets or exceeds a stated goal. If the goal number is 4, what is your probability of winning in your first draw?

expected answer is 0.99984

Answer 1

$P\text{(win game}) = 1 - P(\text{not win game})$

There is only one combination where player doesn't win the game $= P(\text{Ace + Ace + Ace})$

Suits don't matter. So, Combinations of $(A,A,A)= \binom{4}{3} = 4 $
All possible combination $= {52 \choose 3} = 22100$

$$\ P(A,A,A) = \frac{4}{22100}$$

$P(\text{win}) = 1 - 0.00018 = 0.99982$