What is the probability you will pick 4 black cards? with replacement? Without replacement?

Question

You have three picks from a deck of 52 cards. What is the probability you will pick 4 black cards? with replacement? Without replacement?

Answer 1

Probability of 1 black card on 1st draw is 1/2. Half the deck is either spades or clubs. 2nd draw, probability of a black card is 25/51. 3rd draw is 24/50, 4th draw 23/49

Multiply (1/2)(25/51)(24/50)(23/49) for odds of getting 4 black in 4 draws.

Other than saying getting 4 in less than 4 draws is impossible, as odds are zero of getting 4 black in 3 draws, maybe you meant to ask what are the odds of 3 black in 4 draws?

getting 4 out of 3 is like getting blood out of a turnip. Sending a homeless person a tax bill for a billion dollars. Or Biden doubling capital gains taxes and expecting greater tax revenues. Even private companies know there is a limit to prices beyond which revenues and profits decline. but the assumption of profit maximizing firms has a counterpart, the government taxation maximizing tax rate. Raise the rate high enough, is like expecting to get 4 black out of 3 draws. It just doesn't happen.

3 out of 4 though is very possible. 1st draw chances: 1/2, 2nd 25/53, 3rd 24/52, 4th 28/52. Or calculate the odds of BBBN, BBNB, BNBB, NBBB, then sum them.

(1/2)(25/51)(24/50)(28/49) + (1/2)(25/51)(28/50)(24/49)+ (1/2)(27/51)(25/50)(24/49) + (1/2)(26/51)(25/50)(24/49)