How does probability constitute as knowledge in justified belief theory?

Question

This is not the classic lottery paradox. Details of that are available at Epistemic Paradoxes (Stanford Encyclopedia of Philosophy

Suppose there is one lottery with 100,000 tickets and one prize. I have bought one ticket. I know that the probability of each ticket winning is 1:100,000, that the probability of each ticket losing is 99,999:100,00, and that after the draw each ticket will have definitively won or lost.

My rational expectation has to be that my ticket will lose, so buying a ticket seems irrational and if I knew it would lose, I would not buy it.

However, buying a ticket can be justified because the odds of a small loss are balanced against a large gain. There’s no objective criterion for that balance, so buying a ticket is not rational, but also not irrational.

Here’s the paradox:-

1. If my ticket loses, my expectation is true and justified. So it appears to be knowledge.

2. If my ticket wins, my balanced decision is true and justified. So it also appears to be knowledge.

This is not a direct contradiction, but it does seem odd that after the draw, I will have known the outcome whatever it is.

There are three possibilities: -

1. We could accept that in some cases justified true belief is not knowledge.

2. We could adjust the definition of justification.

3. We could adjust the definition of knowledge.

How can this be resolved?

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EDIT AFTER READING ANSWERS SO FAR

At present my position is that: -

1. Probability statements can be known.

2. Probability statements can never justify claims to know unqualified statements about single events. But they can justify beliefs about them and those beliefs may be true.

3. Probability statements can justify decisions to act but only in combination with a calculation or estimate of expected utility and therefore only in the context of the person expecting the utility.

I'm looking for critical evaluation of this position.

Answer 1

There is no paradox here. What you knew for certain before the lottery was that there were 100,000 possible outcomes, in only one of which you would win. The result of the lottery does not change the truth of the knowledge you had beforehand. Suppose that you win- that does not not invalidate your earlier expectation that you would most likely lose. Suppose you lose- that does not invalidate your earlier willingness to place a small bet on the remote possibility of a big win.

Answer 2

In a lottery, the expectation that you will definitely lose is not justified. The expectation that you are very likely to lose with a tiny chance of a huge win is justified.

Now the fact that you will have knowledge in the future that could help you today is useless. Fact is you don’t have that knowledge know, so whether your decision to play or not to play is a good decision, that is independent of the actual future outcome.

Say you are drunk. You are also hungry so you think about driving totally drunk to a fast food place through heavy traffic. Unbeknownst to you there is a fatal failure in your homes gas supply; five minutes from now your home will explode and kill everyone inside.

Driving to get food is the wrong decision. Even though that wrong decision will save your life, it is still the wrong decision.

Back to the lottery: For one dollar you can buy the hope that you might be a rich man by the end of the week, and that hope is what the lottery company sells. And you buy the emotional high when you watch the lottery draw and check your ticket. Is the hope, plus the emotional high, plus the chance of winning, worth the dollar for the ticket? You decide.

Answer 3

It is irrational to buy a lottery ticket, not simply because the chance of winning are low. The expected value of the money you gain is negative. In other words, if you had enough money to buy all tickets to ensure the win, you would lose money. If you buy one ticket for every draw for the rest of eternity, you would (with probability approaching 1) lose money.

This is however only true if you value losing money and winning money the same.

If you value losing money more than you do winning money, it becomes even more irrational.

If you value winning more than you do losing, it might become rational to play until you win, depending on how much more you value the win.

Answer 4

I think that if you speak strictly in terms of probability, point 2 is wrong and buying the ticket is an irrational decision. If your probabilistic model is what you describe, then the odds are extremely poor and if you end up winning it's just luck, you knew you wouldn't win, according to that model.

However, if you want to model the potential gain and the risk of buying a ticket, then probability theory is not enough and you have to switch to decision theory, where you can model the expected utility of either decision (buy a ticket or not) with an influence diagram, and there, depending on your model, you would arrive at the solution with the best utility, that is the most rational one under your modelling constraints. But these are augmentations of standard probability theory that take into account extra contextual information such as the economic gains you mention. In fact, methods for maximizing the expected utility (such as the influence diagram) have been pretty used in economics.

Answer 5

You are making an assumption that it is irrational to play the lottery when you say that it makes no sense to buy something with a low probability. Then, you say, that because the gain is really high, the minimal cost is worth it. However, the latter case would still be a situation where you have a low probability of winning. This creates a contradiction in your scenario from the out go.

Either it is rational to bet on things even with low probabilities or it is rational to do so in some cases. Now, the question of whether or not it is rational is ultimately subjective. It also depends on the context. A person who has tons of money lying around may argue that it is rational to play lotteries since he does not have much to lose.

Ultimately, it depends on what you define as rational. As long as that definition is coherent, there is no contradiction.

Answer 6

The fallacy is thinking that if you win, your decision to buy a ticket was true and justified (or the best decision of their live as some would say it). If you win, you should think: considering the expected value, I was underpaid.

This is how good poker players can still improve after making a bad bet, even if they win. They are able to reflect on their decision and calculate if their win matches the risk they took and avoid that mistake in the future. Bad players feel their strategy was validated and will repeat that mistake even after continuing to lose.

The difference in Lottery is of course that you likely will play much fewer games in life than you could play poker hands. But collectively, all participants play enough games to apply probability theory. It's just that the bad decision is shared among them (including the winner).

Answer 7

My rational expectation has to be that my ticket will lose, so buying a ticket seems irrational and if I knew it would lose, I would not buy it.

However, buying a ticket can be justified because the odds of a small loss are balanced against a large gain. There’s no objective criterion for that balance, so buying a ticket is not rational, but also not irrational.

You argue in your 2nd statement that there cannot be an objective criterion for that balance yet at the same time argue in your first statement that it's almost knowledge that you will lose and that it's irrational to do so? Why?

Like if it was just about how much money you make then there is an objective criterion to measure that balance: the expectation value. Meaning in the limit of infinite games the 1:100,000 ratio becomes reliable and so you can calculate jackpot-100,000 cost of ticket = net win/loss after 100,000 games. Divide that by 100,000 and you end up with the expected return per ticket.

As most lotteries only give back 2/3 of the ticket prices or less, that per game return is almost definitely negative. So regardless of winning or losing on average you'll lose money playing the lottery.

So if you look at the lottery as a reliable source of income, that would be irrational, because reliable it's only a source of income for the host of the game not the player.

There could be other motivations like thrill, hope, temporal high or other precursors to a gambling addiction. But in that case winning or losing might become a secondary motivation.

But you could also imagine the scenario that you owe money to the wrong people who vow to kill you tomorrow if you don't give them the money. But the money that you got is far less than expected and to little to run away, but enough to buy a few lottery tickets. So the odds of the lottery saving you are vanishingly low but so are your odds of survival in general and dead that money you have left is without use to you.

Now these are the ideal costumers for a lottery because they look for luck as a last resort and doing so makes them even more dependent on luck (as on average it's a money drain not a source).

Nevertheless it might be rational to buy the ticket because a low probability is still a positive probability.

TL;DR your choices in the context of statement 1 and 2 are not independent. If you think the balance is worth it, than 1 is not irrational despite the low odds. And if you think that 1 is not worth it because of the low odds than the balance in 2 is also negative.

Like even if the valuation is subjective it's likely still consistent for any given individual and you seem to apply a more general assumption on 1 but an individual for 2 which is likely not how this works.

Also the outcome of a single probabilistic event cannot be known or justifiably believed. So seeing the result doesn't change the knowledge (unless you believed it's impossible/inevitable and were wrong about that). If it was unlikely to begin with, you'll still see it as luck if you win and wouldn't be surprised if you lose. It only becomes a certainty in the limit of infinite trials of a fair game. In that case you'd gain knowledge about the probability distribution and the games expectation value. Though you'd still be clueless about an individual game, yet you could estimate if the odds are in your favor or against you.

Answer 8

You are quite simply mixing up propositions (i.e., general knowledge about how lotteries work and how likely it is to win versus the concrete result of a single draw) in the layout of your argument.

Only because you know whether you won or lost after the draw, does not mean that you buying a ticket beforehand meant that you somehow "knew" that you would win (and were rational) or that you somehow knew that you would lose (and thus behaved irrationally).

This has nothing specific to do with lotteries but goes for anything that required an experiment or observation. Let's say I wish to know how many cars pass my street today. I can just guess a number. Then I can sit all day and count the cars. Me guessing the right (or wrong) number says nothing whatsoever about me "knowing" beforehand what the result would be. But back to your question.

To be more formal, for something to count as knowledge, you need some aspects, described e.g. on the Gettier Problem page of Wikipedia:

A subject S knows that a proposition P is true if and only if:

• P is true, and
• S believes that P is true, and
• S is justified in believing that P is true

In your case, the fact of whether you guessed the right number of cars, or whether your ticket one, is just the first bullet point. The second bullet point, in your example, is of little importance (you do not assume to know which ticket will win before the draw). The third one is the jackpot. Even if you believe that ticket #6654 will win, and even if #6654 eventually is drawn, at no point could you claim that you were rationally justified that this would be the case (assuming it is a normal lottery which always has a negative expectation value). So no matter what you do, you would not count the fact that you won as "knowledge" that you somehow had beforehand.

Finally, all of this has nothing to do with whether it is rational or irrational to buy a ticket - this would depend on many factors (i.e., if the prize money becomes excessively large, then mathematically the expectation value could become positive, and there would come a point where it would be wise to play on the off-chance that you're the winner; or if you draw a lot of pleasure out of playing, even though you knew you would never win, and certainly wouldn't win back your money, it could still simply count as entertainment and could be worthwhile if you judge it so).

Answer 9

Probability statements can be known.

This is true. It's the same thing as knowing the shape of the die being tossed. How many sides it has. It is not the same as knowing how it will land.

Probability statements can never justify claims to know unqualified statements about single events. But they can justify beliefs about them and those beliefs may be true.

This is known as anecdotal evidence. "I lost 50 pounds eating nothing but rutabaga for a week" may be true for them but that doesn't mean it will work for you.

Probability statements can justify decisions to act but only in combination with a calculation or estimate of expected utility and therefore only in the context of the person expecting the utility.

Oh you need much more than that. For one you need a fair game.

A classic game is the Monty Hall Problem. You have to choose one of three doors. Behind 1 is a valuable prize. Behind the other two are cheep stuffed goats. When you chose a door another one is opened and you're shown a goat. Now you're offered a chance to switch doors or stick with the one you picked.

Unsuspecting mathematicians will tell you it's always better to switch. It gives you 1 in 2 odds. Staying only gives you 1 in 3. So your choice is clear.

When you're sure you aren't being scammed.

What the game shows you is that the one conducting it knows exactly where the prize is. What you don't know is if you are only being given the chance to switch because you already picked the valuable prize. You can preach about the odds all you like but this is really about trust. Trust me when you shouldn't and I can stick you with a goat every time. And since I never told you we were playing Monty Hall I'm not even cheating.

If my ticket loses, my expectation is true and justified. So it appears to be knowledge.

All you know is that you lost.

If my ticket wins, my balanced decision is true and justified. So it also appears to be knowledge.

All you know is that you won.

This is not a direct contradiction, but it does seem odd that after the draw, I will have known the outcome whatever it is.

But what you don't know is why that outcome happened. It could be that your great uncle Stanly wants to give you 10 million dollars but doesn't want you to know. So he set up a fake lottery and got someone to sucker you into playing.

Outcomes only teach us when the game isn't rigged. Science fanatically checks against bias for exactly this reason. If the game is fair the law of large numbers takes over once you have enough trials, that is, outcomes to have reasonable confidence that your data is really telling you something you can trust. So long as no one has their thumb on the scale.

Even when the game is fair, outcomes only teach us when we do the math right. When we construct a model correctly. Get unlucky enough you can confirm a bad theory that you would have disproven if you'd have just thrown the die one more time. Which will be true no mater how many times you throw it.

One outcome only teaches you that that outcome was possible. Not exactly how likely. And only if you know what you're looking at.

We could accept that in some cases justified true belief is not knowledge.

I think therefore I am. For everything else I just hope I'm not worth the effort to fool.

We could adjust the definition of justification.

OK sure. I can justify playing the lottery regardless of the odds. I pay $1, buy a ticket, lose, and decide that was fun. Fun enough to buy another ticket. So long as no one tells me the game was so rigged I was never going to win, regardless of how lucky I got, it doesn't matter. I'm still having fun. Blissfully ignorant fun.

That's why people play the lottery. It is rational. It just has nothing to do with the math. Which is how they get ya.

Since the odds of winning the lottery are so low, and you have to get so lucky, lets use that imagined luck to give us not only a lot of money but the best lottery winning story ever.

We can use the same justification to play a different lottery. I'm playing it right now. Hold out your hand. Imagine a winning lottery ticket falling into your hand. Now snap your fingers and open your hand. Did a ticket fall into it? No? Well sorry you lose. But if you had fun it was worth it and cost you nothing but a moment of time thinking positive thoughts. It's the You-Don't-Have-to-be-In-It-to-Win-It Lottery^tm. You played it just by reading this.

We could adjust the definition of knowledge.

Be careful with this. Our brains are pattern matching machines that can see rubber ducks in clouds.

This is not a direct contradiction, but it does seem odd that after the draw, I will have known the outcome whatever it is.

No, you don't really. Because a flipped coin can land heads, tails, on edge, or roll into a storm drain. However weird you think the universe is, it's weirder. But since we're not dead there must be some normal hiding somewhere.

Answer 10

The notion of probability is not well defined in purely physically terms, this is an unsolved philosophical problem as pointed out by David Deutsch here. There is no known way to rigorously define probability in terms of physically realizable systems, all attempts end up invoking probability in a self-referential way.

Answer 11

In this statement:

If my ticket loses, my expectation is true and justified. So it appears to be knowledge.

The "knowledge" comprises your expectation combined with the observation, after the fact, that the ticket did lose. Prior to the lottery draw, all you had was a guess; afterwards you have empirical knowledge about the winning ticket, plus a guess.

If you are considering this from an outside viewpoint -- an omniscient being, say, or a person reading about it in a novel -- then there may be other things at play. In that case the gambler's expectation could be foreshadowing, or it could affect the narrative in some magical way.

But if it's you buying the ticket, then what constitutes "knowledge" is constrained by the way your subjective universe works. Your thoughts about the ticket are thoughts about a hypothetical, fictional future world, and whether your guess proves correct or not, the imaginary world you knew about remains distinct from the real world where the draw took place.

Your mental model parallels the real world to the extent that you can "know" the draw will happen on Saturday. Granted, this is a probabilistic kind of knowledge too, since the machine could always be hit by a comet. Indeed, there are ways you could be wrong about what color your own eyes are. But the categorical difference between knowing you're unlikely to win the lottery, and knowing you won't win, is that the latter stipulates a specific outcome; until you know what ticket will win, you don't know it won't be yours.

Answer 12

You totally miss the point.

We play in the lottery for amusement, not financial profit.

The potential financial profit adds some sort of thrill. Some people are more addicted to than others. The "profit" is thrill and excitement; the cost is (usually) rather small, so the thrill is not overshadowed by great risk.

While your positions are indeed valid, the assumption that rational thinking is the only driving force is wrong.

Or you could simply say that lotteries exploit a human weakness where our mind suspends rational thinking (naturally, there are human who are more vulnerable to lotteries than others). Knowledge of that human weakness makes a lottery rational for the lottery institution which can predict a financial profit.