Faby took the four cards with rank J, (Jacks) the four cards with rank Q,(Queens) and the four cards with rank K(Kings)from a standard deck of cards, shuffled them, and then picked a first card and a second card (without replacement).
The probability that the first card she picked has a lower rank than the second card is P/Q as a fraction in lowest terms. What is P+Q?
+6251 \(P = P[Q \cup J | K]P[K]+ P[J|Q]P[Q]\\ P=\dfrac{8}{11}\dfrac 1 3 + \dfrac{4}{11}\dfrac 1 3 = \dfrac{12}{33} = \dfrac{4}{11}\)
Another way we can do this is to note that the number of pairs where card 2 > card 1
is equal to the number of pairs where card 1 > card 2 and that these two plus the number
of pairs where cards 1 and 2 are equal equal the total number of pairs.
\(P[\text{card 1 = card 2}] = 3 \cdot \dfrac 1 3 \cdot \dfrac{3}{11} = \dfrac{3}{11}\\ 1 = \dfrac{3}{11} + 2 P\\ 2P = \dfrac{8}{11}\\ P=\dfrac{4}{11}\)
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