So let's say I roll a 4 sided die. Whatever number it lands on is the number of cards I have to draw from a standard 52 card deck. So if it lands on a 3 then I have to pick up 3 cards. I win if I get a heart. What is the probability of not winning after I play once? There's 13 hearts out of 52 cards so I assume I have a 0.25 probability of drawing a heart but how can I use the dice roll to figure out what my chances are of losing the game on my first try?
[Math] Finding the probability of drawing a heart from a deck of cards.
probability
Best Answer
There are four possibilities for your dice roll.
These possibilities are mutually exclusive and exhaustive.
We also need the probability of drawing $n$ cards and not getting a heart. For 1 card this will be $(52-13)/52$, for two cards this will be $\frac{(52-13)}{52}\cdot \frac{(51-13)}{51} = \frac{39\cdot38}{52\cdot51}$, etc.
So for $n$ cards this will be $$\frac{\prod_{i=1}^{n}(40-i)}{\prod_{i=1}^{n}(53-i)}$$
So now our probability is simply the sum of the probabilities of the dice rolls multiplied by their corresponding card drawing probabilities, which are all $\frac{1}{4}$.
$$\frac{1}{4}\sum_{k=1}^4\bigg[ \frac{\prod_{i=1}^{k}(40-i)}{\prod_{i=1}^{k}(53-i)}\bigg] = \frac{1}{4}\sum_{k=1}^4\prod_{i=1}^{k}\bigg[ \frac{(40-i)}{(53-i)}\bigg]$$