SOLUTION: you draw two cards at random from a standard 52-card deck. What is the probability that the first card is a 2 and the second card is a 3?

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Question 769273: you draw two cards at random from a standard 52-card deck. What is the probability that the first card is a 2 and the second card is a 3?
Answer by DrBeeee(684) About Me  (Show Source):
You can put this solution on YOUR website!
When you say draw two cards from the deck, it means you keep them out of the deck. This is drawing without replacement. The solution is straight forward. There are four twos (deuces) in a standard deck of 52 cards, so the probability of drawing a two on the first draw is
(1) P(2) = 4/52, OK?
Now you keep that 2 in your hand so there are only 51 cards left in the deck, four of which are threes. Now to draw a three is
(2) P(3) = 4/51 (We would divide by 52 if we replaced the first card)
The compound event of drawing a 2 then a 3 without replacement is given by the product of the two independent events or
(3) P(2 then 3) = P(2)*P(3/2) where P(3/2) reads as the probability of getting a 3 after we got a 2. or
(4) P(2 then 3) = (4/52)*(4/51) or
(5) P(2 then 3) = 4/663
It's important to note that I specified that the two event draw a 2 then draw a 3 are INDEPENDENT. Sometimes they both may occur together. In that case they are not independent and we use the following calculation for two events, A and B
(5) P(A then B) = P(A)*P(B/A) - P(A and B)
Clearly this does not apply here because we can't, in a single drawing get a 2 and a 3.
Answer: After removing two cards from a standard deck of cards, the probability that the first card is a 2 and the second card is a 3 is 4/663.
PS It doesn't matter whether or not you draw two cards at a time or one card at a time. Only if we replace the "first" one or not.
PPS The above calculation assumes that the first event (or all previous events) occurred favorably, ie in this problem a 2 was drawn on the first draw. For example if you want the probability of getting 2 then a 2, the answer is (4/52)*(3/51), OK? For 2 then a 2 then a 2 the answer is (4/52)*(3/51)*(2/50) Get it?