How do I determine the probability I will draw three or more cards with a value of 8 or higher from a standard deck of cards when drawing five cards?

Question

Jokers are not in the deck, aces are value 1, jacks are value 11, queens are value 12, and kings are value 13.

Answer 1

Assuming that a drawn card is replaced in the pack before the next card is drawn, the probability of drawing a card with value of 8 or higher is 6/13.

The probability of a value less than 8 is 7/13.

Let p=6/13, q=7/13.

The binomial for this probability is (p+q)⁵=p⁵+5p⁴q+10p³q²+10p²q³+5pq⁴+q⁵=1.

For 3 or more cards with value≥8, we can calculate the total probability:

p⁵+5p⁴q+10p³q²=0.4282 approx, 42.82%. That is, the sum of all≥8, 4 cards≥8, 3 cards≥8.