Chances of A or K (8 outs) being dealt by river, not taking into account what we hold in our hand…..

Flop:
8/52 = 15.38%
8/51 = 15.68%
8/50 = 16%

Turn:
8/49 = 16.32%

River:
8/48 = 16.66%

Total = 80.04% Chance of an A or K being dealt by the river.

Is this math correct?

Thanks

You wont draw another card on river so the 16.66% should not be there. I dont know if the rest of it, cca 64% is correct tho

No, you don't add probabilities like that. For example if you used your calculation for 7 cards rather than 5, the probability would exceed 100% which is clearly wrong.

The best way to solve this particular problem is to look at the reverse problem (what is the chance of no ace or king by the river) and subtract it from 100%.

1st card chance of no ace/king = 44/52
2nd card chance of no ace/king = 43/51
...
5th card chance of no ace/king = 40/48

Then the probability an ace or king = 1 - (44/52)*(43/51)*(42/50)*(41/49)*(40/48)

= 58.21%

I meant if all 5 cards are dealt, complete runout.....

Thanks guys!!! Got it. This question arises from a conversation I had with a fellow I work with who used to play online. He had tracked about 1000 hands and had said that he found an A or K had come at least 64% of the time in full run outs. After we had the correct calculation I assumed this had allot to do with the small sample size as opposed to a rigged RNG which he said was possibly in play. LOL. Love 2+2 post a question in Theory and get an answer in under 30minutes . Have a great weekend boys!!!


cheers

The observed 64% is a bit surprising simply because if a hand went to completion, there's a pretty good chance that at least one player has an ace or king, making the theoretical probability even smaller than 58%.

The chance of observing an Ace or King 640 or more times in 1000 hands is about 0.011% given we accept the optimistic 58% number. So, we can either suspect rigging or faulty data analysis such as a wrong filter. I strongly go for the latter.

An alternative way to calculate this would be 1-(runouts without an A/K)/(total runouts), or 1-(44 choose 5)/(52 choose 5) = 58%

I think a vast majority of these types of situations arise as a result of selection bias. Someone running bad is looking for a reason that the game isn't fair and of course they will find something. If everybody was posting about their experience of seeing too many As/Ks, or getting oversetted too much, or getting AA cracked AIPF too much, it would be suspicious, but instead each individual has distinct types of run-bad most of the time.

Even though the event of getting so many As/Ks on 5-card runouts is pretty low, the pool of events being chosen from was likely enormous. What I mean by that is, if a normal amount of As/Ks appeared, we'd likely instead be asked the probability of not getting AA once in 1000 hands after it happened to the asker

I suspect humans are in fact more likely to continue on flops NOT containing an ace or king, thus the extra runnouts WITH an ace or king since the deck had a few extra.

Run the data on flops only. Should be much closer to the correct percent, within margin of error.

Are we holding anything or not? (52 cards live)