Blacjack dealer has a rare hand question

[pokerscab1]

Was dealing last night on table, 6 deck shoe. I turned up a four, flipped over a 4, then pulled three more fours, much to MY delight, but to the players dismay. (I thought it was neat.)
I've been dealing for 4 years and have only seen this one time. Was wondering what the odds of dealing this hand are.

[gaming_mouse]

Interesting...

First note that we can assume your cards are always dealt in order. That is, without the gaps between the first, second, and third card that result from a typical blackjack deal. Never mind how we do this -- maybe by creating two piles from the shuffled deck, one for players and one for the dealer -- the point is that the assumption will not affect our calculation.

So, first the deck has to contain 5 4's in a row. Let's calculate the chance of that. 312 cards in the shoe, 24 fours. 312! possible shuffles. Since you probably only deal about half the deck, the sequence of fours (to be playable) can begin anywhere in the first 160 positions, let's say. Thus the number of possible shuffles which contain a playable sequence of at least 5 fours is:

160*(24 choose 5)*307!

So the odds of having a shuffled a shoe where this is even possible are:

160*(24 choose 5)/(308*309*310*311*312)= 2.37554856 × 10-06, or .00000237 -- about 1 in 416,000.

But we're not done. Because the sequence has to begin at a point in the shoe where a dealer hand begins. Now the number of players you are up against comes in. Against one player, you are more likely to begin your own hand on the beginning of the sequence. Let's say you were up against 3 players though, and that the average BJ hand contains 3 cards (I'm just guessing -- you may wish to adjust this). Then your own hand begins every 16 cards on average. So we need to divide the above number by 16 -- which means brings up to a little over 6.5 million to 1.

Finally, we note that you would have been equally surprised by 5 aces, twos, or threes, so we divide by 4, since these events are (for practical purposes) essentially mutually exclusive.

This brings us back down to about 1.6 million to 1.

Note the crucial estimations: % of shoe you are dealing, and number of cards in the avg BJ hand, which I guessed at. Changing those number will change the final result, though we are in the right ballpark I'm sure.

[pokerscab1]

Thanks for the info. Kinda puts it all into perspective about how tough it is to win the powerball jackpot. I deal 400 hands per hour, so one could say I buy 400 tickets per hour, 8 hours a day, 5 days a week, and I still only beat 1 in 6.5 million odds once in 4 years. What are powerball odds? 1 in 175 million? I guess the dream is worth the $1.00. Thanks again.