Two cards are drawn successively without replacement from a well-shuffled deck of 52 cards. Find the probability of exactly one ace.
Given that two cards are drawn from a well-shuffled deck of 52 cards.
We know that there will 4 aces present in a deck.
It is told that two cards successively without replacement.
Let us find the probability of drawing the cards.
⇒ P(A1) = P(Drawing ace from 52 cards deck)
⇒
⇒
⇒ P(O1) = P(Drawing cards other than ace from 52 cards deck)
⇒
⇒
⇒ P(A2) = P(Drawing ace from remaining 51 cards deck)
⇒
⇒
⇒ P(O2) = P(Drawing a card other than ace from remaining 51 cards deck)
⇒
⇒
We need to find the probability of drawing exactly one ace
⇒ P(DA) = P(Drawing exactly 1 ace in the drawn two cards)
⇒ P(DA) = P(Drawing Ace first and others next) + (P(Drawing Other cards first and ace next)
Since drawing cards are independent their probabilities multiply each other,
⇒ P(DA) = (P(A1)P(O2)) + (P(O1)P(A2))
⇒
⇒
⇒
∴ The required probability is .