Lottery: If two options of lottery tickets are available to you both with a chance at winning $100 but either has an additional stipulation unknown to you then the chances of you winning any money at all is 50/50.
Say Ticket A's additional piece of information is that even though the statistical odds of any given person winning $100 are 25%(I don't like to do math with 30%) every 10th winning ticket has a multiplied payoff of x4 winning you $400 meaning you actually don't win $100. That said it is completely exempt from being reveled within the statistical odds of 25%. The other has the 75% for $100(adjusted to keep the odds three times as high that you get a hundred dollars with the changed 30%->25%) But the hidden information on this ticket is that every 4th winning ticket holder is responsible for offsetting the tax on all transactions between the company and customer. This would mean that even if you win $100 there is a chance that you'll win it and then be put into a select few who are responsible for using a portion of their winnings, lets say $70 for each fourth winning ticket, to make the system cheaper for the rest of the actors involved. This means that even though you are one of the 75% of winning ticket holders you actually only won $30 and thats before the cost of the ticket itself. In this way someone who says logically that there is a 25% chance of winning up to $100 depending on which ticket you chose would be right based on only knowing there are two tickets and either could be a winning or a losing ticket. When you take into account that the tickets are not supplying you any false information by saying "One in four wins up to $400!" or "Odds are 3 to 1 for a $100 payoff!" just withholding information then the power of probability only comes into account once you have that extra information and the populous does not. In that case your logical deduction might be something like: ticket a has a 1 in 4 chance of winning every 10th win rolls giving me a likely chance of winning twice in a row and a bonus 400. I buy 20 tickets then, I win 5 $100 tickets and one $400 which is pretty much aligned with my information and I unfortunately just missed out on the second high priority win. Still $900 and if we say each ticket cost $20 with a 20% tax on lottery tickets in this world then my tickets individually cost me $22 and that 20 times is $440 meaning I still won net $460. The other ticket with a 75% chance of winning would yield 15 wins and $100 each there with 5 unlucky tickets meaning I picked up an extra toxic ticket again within reason of expectation. $100 15 times is $1500 then $70 five times is $350 which brings me down to $1150 then I pay $20 for each ticket $400 in total which means I won a grand total of $750 this means that for someone who can spend $400 on lottery tickets they could find this information to realize its way better to play through the high chance wins rather than the jackpot goal but you can see how the information hidden could easily sway the "logical probability" without your knowledge.