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The answer depends on how many cards are in the hand.
Approximately 2%
The probability of five cards being four cards from one suit and one card from another suit is the same as the probability of drawing four cards from one suit multiplied by the probability of drawing one card from another suit, multiplied by 5 (for each of the possible positions this other card can be drawn in). The probability of drawing four cards from one suit is 12/51 x 11/50 x 10/49. The probability of drawing a fifth card from another suit is 39/48. All these numbers multiplied together (and multiplied by 5) come to 0.0429. So the probability of drawing a hand of five cards with four cards from one suit and one card from another is 5.29%
Total combinations for drawing 5 cards is 52 choose 5 which is...52*51*50*49*48/(5*4*3*2*1) possible outcomes for drawing 5 hearts is 13 choose 5 which is...13*12*11*10*9/(5*4*3*2*1) Probability of drawing 5 hearts is (52 choose 5)/(13 choose 5) which is....000495
Since it's video poker, we can assume that the only cards drawn are the ones you started with. There are exactly 47 cards left in the deck. There are 6 series of draws you can get that will give you ace, queen, and 10 (3!, or 3x2x1).The number of sequences of 3 cards in a 47 card deck is 47!/44!, or 47x46x45, and since 6 of the sequences leave you with the hand you want, you have exactly 6/(47x46x45) probability to get one of them.This works out to 1 in 16,215.