Part 1
The smallest number of cards you'll have to draw will be 1 card, and there are 13 hearts out of a 52 card deck
13/52 = 1/4 = 0.25 or 25% probability of drawing a heart.
Part 2
The largest number of cards will be drawing every other suite before you stumble on the heart, so you draw all 39 cards, then 1 more for the heart, so you have to draw 40 cards total.
Since they are being drawn without replacement, we multiply the probability of drawing each one individually...
1 - (40/52) • (39/51) • (38/50) • ... (1/13)
[The difference between denominator and numerator is constant, 52 - 40 = 12, 51 - 39 = 12, 13 - 1 = 12, hence how we end up with 1/13]
2 - To formulate this pattern, the top fraction goes from 40 to 1, which is 40!
3 - 52 stops at 13, so we can't simply divide by 40!/52!, but luckily there is a formula for truncating 52! at 13 in the denominator of 1.
n • (n-1) • (n-2) ... [n - (k-1)] = n! / (n - k)!
52 • 51 • 50 ... [52 - (39)], where k-1 = 39
k = 40
∴ n! / (n - k)!
= 52! / (52 - 40)!
= 52!/12!
Combining part 1, 2, and 3
Therefore, the odds of drawing all other cards and then your heart will be:
= 40! / (52!/12!)
= 4.85 x 10-12 is the probability of drawing 40 cards in order to draw a heart.
