Ask the Wizard #147

No. The probabilities do not change in any card gave by burning cards.

There is only one way to get that exact hand. So the probability would be 1 in combin(52,5) or 1 in 2,598,960.

The expected return on the straight bets would be (0.5*1 + 0.5*(-1.2))/1.2 = -8.33%. The expected return on the parlay would be 0.25*2.5 + 0.75*-1 = -12.5%. However if I were to only bet two games and want to win or go bust trying then I would go with the parlay. More importantly I would boycott this bookie out of principle, because I’ve never heard of having to lay -120 on straight bets before.

Thanks for the kind words but I'm not that smart. A couple years ago I took the Mensa entrance exam, and didn't make the requisite top 2%. I'm still upset that they refused to tell me how well I did do. On January 13 Jeopardy tryouts are coming to Vegas, for which I have an appointment, and am sure I'll blow that too. Anyway, to answer your question here you go:

With suited hole cards:

Flush after flop: combin(11,3)/combin(50,3) = 165/19600 = 0.842%.
Flush after turn: (combin(11,2)*39/combin(50,3))*(9/47) = 2.096%.
Flush after river: (combin(11,2)*combin(39,2)/combin(50,4))*(9/46) = 3.462%.

With unsuited hole cards:

Flush after flop: 0%
Flush after turn: 2*combin(12,4)/combin(50,4) = 0.430%.
Flush after river: (2*combin(12,3)*39/combin(50,4))*(9/46) = 1.458%.

Here are the commulative probabilities.

With suited hole cards:

Flush by flop: 0.842%.
Flush by turn: 2.937%.
Flush by river: 6.400%.

With unsuited hole cards:

Flush by flop: 0.000%
Flush by turn: 0.430%.
Flush by river: 1.888%.

Birthdays are such relationship killers. If he isn’t blatantly cheating he is at least hedging his bets by heating up things with those on the waiting list. However I can’t say that I blame him because you seem paranoid and possessive. My advice is to lower the temperature on this. Do as he does and heat up some friendships with other guys as a back up plan. Either he will get jealous and make a stronger effort or it will hasten the eventual ending, which are both better than continuing to go sideways.

That shouldn’t change the odds at all. The Windows RNG is probably not very good, but good enough for free play. However when real money is on the line a smart operation would use a proven good RNG on their own end.

That would be $35 above average, or 7/9 standard deviations. The probability of being more than 7/9 standard deviations above expectations would be 1-Z(7/9) = 1- 0.78165 = 0.21835.

I would put those behind it on the same level as those selling get rich quick gambling schemes. The mathematically ignorant taking advantage of the mathematically ignorant.

I would ask your ex what his evidence is. Accusations should always be backed up with evidence. Maybe there is suddenly some bad blood between the two of them and this is a way of your ex seeking retribution. It seems unlikely he would make up this story out of thin air so there may be some basis in truth to it. However he is professing more knowledge than I think it totally believable. So probe him for more details.

Player B would need to win the next two (not counting ties) so the probability is (1/2)*(1/2) = 1/4.

The probability of a random shuffle resulting in starting order is 1 in 52!, or 1 in 8.06582*10⁶⁷. If you did a perfect shuffle, in which last card was the first to come down, thus remaining last, it would only take 8 shuffles to be back to the starting order. If the 26th card was the first two come down then it would take 72 shuffles to back to the starting order.