i'm confused by the following problem could someone walk me through it, so i can understand
(a) 10 cards are drawn at random one at a time with replacement from an ordinary deck of cards.
- What is the sample space?
- What is the probability that no Ace appears on any of the draws?
- What is the probability that at least one King appears in 10 draws?
- What is the probability that at least 2 Queens appear in the 10 draws?
(b) What are the corresponding probabilities in (a) if the drawing is done without replacement?
Solution: (a) The sample space is a sequence of ten cards, and its size is 5210
No Ace has probability (48/52)10
By looking at the complement, at least one King has probability 1-(48/52)10
There's a (10 choose 1)(4/52)(48/52)9 probability of getting exactly one queen, so by compliment at least 2 Queens has probability 1 - (48/52)10 - (10 choose 1)(4/52)(48/52)9
(b) Since we are drawing without replacement, it is easier to think of drawing (un-ordered) subsets, and the sample space is all hands of ten cards;
Thus, no Ace has probability (48 choose 10)/(52 choose 10)
At least one King has probability 1 - [(48 choose 10)/(52 choose 10)]
At least 2 Queens has probability 1 - [((48 choose 10)+4*(48 choose 9))/(52 choose 10)]