Solution
The Plain Straight
Show the problem again
A 5-card hand is dealt uniformly at random from a standard 52-card deck. A straight is five cards of consecutive ranks. The ace may play high (10-J-Q-K-A) or low (A-2-3-4-5) but does not wrap around (Q-K-A-2-3 is not a straight). What is the probability that the hand is a straight but not a straight flush, so its five cards are not all of the same suit? Give your answer as a reduced fraction.
Worked solution
The answer is 5/1274. A straight is determined by its lowest card, which ranges from A to 10, giving 10 rank patterns. Each of the five cards can be any of 4 suits, giving 4^5 = 1024 suit choices, of which 4 are monochromatic (straight flushes). That leaves 10 × 1020 = 10,200 hands. Dividing by C(52,5) = 2,598,960 gives 10200/2598960 = 5/1274 ≈ 0.39%. A straight is about twice as common as a flush, because requiring one suit costs a factor of about 4^4 while consecutive ranks cost less.
Source: Standard poker-hand probability exercise from introductory probability texts. Statement written for AxiomIQ.