Author

Topic: Judgements of relative likelihood (for fun)

torontoprofessor
rabblerouser
Babbler # 14260

posted 30 July 2008 01:54 PM
Here's an imaginary profile of Linda: Linda is a 31 years old Canadian woman. She is single, outspoken and very bright. She majored in philosophy at university. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antiwar demonstrations.Rank the following ten statements according to their probability, from most likely to least likely. (a) Linda is a teacher in elementary school. (b) Linda works in a bookstore and takes Yoga classes. (c) Linda is a teacher in an elementary school and usually votes Conservative. (d) Linda is active in the feminist movement. (e) Linda is a bank teller. (f) Linda is a psychiatric social worker. (g) Linda usually votes NDP. (h) Linda is a bank teller and is active in the feminist movement. (i) Linda is an insurance salesperson. (j) Linda usually votes Conservative. Example ranking (not necessarily my own): g d a b f h e i j
From: Toronto  Registered: Jun 2007
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M. Spector
rabblerouser
Babbler # 8273

posted 30 July 2008 01:56 PM
I think they are all equally probable.ETA: on second thought, logically c is less likely than a or j, because both a and j would have to be true for c to be true. Similarly, h is less likely than d or e. [ 30 July 2008: Message edited by: M. Spector ] ETA: Ha! I avoided the Conjunction fallacy. [ 01 August 2008: Message edited by: M. Spector ]
From: One millihelen: The amount of beauty required to launch one ship.  Registered: Feb 2005
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Sven
rabblerouser
Babbler # 9972

posted 01 August 2008 02:58 PM
(g) Linda usually votes NDP. (d) Linda is active in the feminist movement. (a) Linda is a teacher in elementary school. (b) Linda works in a bookstore and takes Yoga classes. (c) Linda is a teacher in an elementary school and usually votes Conservative. (i) Linda is an insurance salesperson. (h) Linda is a bank teller and is active in the feminist movement. (e) Linda is a bank teller. (f) Linda is a psychiatric social worker. (j) Linda usually votes Conservative.Reasoning: ■ She is probably progressive (and therefore a relatively higher likelihood of (g) and (d) and a relatively lower likelihood of (j)). ■ I think she is equally likely to be (a) or (b) (because vocationally, those are probably going to be the highest choicesrelative to the other vocational choicesgiven her education) ■ I ranked (c) where I did because, vocationally, she's less likely to be an insurance salesperson or a bankteller than a teacheragain, given her education ■ I ranked (i), (h), and (e) the way I did because being an insurance salesperson is probably more difficult and challenging than being a bank teller (and, if she is a bank teller, I would guess she's more likely to be active in the feminist movement than notthus, (h) before (e)). ■ I ranked (f) low because it doesn't seem to mesh with her university degree.
From: Eleutherophobics of the World...Unite!!!!!  Registered: Jul 2005
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Sven
rabblerouser
Babbler # 9972

posted 02 August 2008 08:37 AM
Thanks for that link, M. Spector."Many people are adverse to taking risks. People tend not to bet $500 on a 50% chance of winning $1,000, even though that is the fair price." I"m not sure what they mean by a "fair price". Statistically, the "expected value" of that bet = (A) 0.50 x $1,000 + (B) 0.50 x zero = $500. Or, to put it another way, the "expected value = (A) 0.50 x $500 + (B) 0.50 x ($500) = zero. If a person was faced with a large number of such bets, the person would, statistically, end up with the same amount of money the person started out with in the beginning. So, it's rational that people would avoid the bet (why go through the effort of entering into the bets when, statistically, you would end up with the exact same amount of money you started with?) In contrast, if the chances of winning $1,000 was anything in excess of 50% (even 50.00000001%) and if the chances of losing was anything less than 50% (even (even 49.99999999%), then I would take every bet given. What's interesting is the "expected values" in casino gambling (like slot machines)and why people would ever gamble. Let's say you put a quarter in a slot machine and you either lose your quarter or you win $100. The casino (in order to make money) has to create an expected value of less than $0.25 per bet (but only slightly less than $0.25 per bet). So, let's say that the odds are set so that you lose, on average, $0.245 per bet. That means that on every bet, your chances of winning $99.75 (getting $100 in exchange for giving up your $0.25 bet) are 1 out of 200 bets (and, conversely, your chances of losing your quarter entirely are 199 out of 200 bets). So, you have a 99.995% chance of losing your quarter and you have a 0.005% chance of walking away with $99.75. Yet, people still do it...and they sit at slot machines and do it for hours upon hours, feeding those friggin' machines with quarters. Gambling (from the gambler's perspective) is for the mathematically ignorant (unless it's something that takes skillperhaps a rare horse better or a person who can count cards and who can quickly and accurately calculate probabilities in their head, such as in blackjack, even with a dealer using four decks). ETA: The formula for my gambling example is: $99.75 x 0.005% + ($0.25) x 99.995% = ($0.245) [ 02 August 2008: Message edited by: Sven ] [ 02 August 2008: Message edited by: Sven ]
From: Eleutherophobics of the World...Unite!!!!!  Registered: Jul 2005
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Sven
rabblerouser
Babbler # 9972

posted 02 August 2008 01:37 PM
quote: Originally posted by M. Spector: I think the meaning is obvious: a "fair price" is one that involves neither a positive nor negative "expected value"  that is, "fair" for both sides.
...and a transaction that neither would bother with because the net result is to (statistically) end up in the exact same place one started in prior to the transaction. quote: Originally posted by M. Spector: If people are willing to bet repeatedly at slot machines with a negative "expected value" (however slight), how can you say that "rational" people would not take the "fair price" bet? Or is that the very point you were making: that slot machine betting is less rational than betting $500 on a 50% chance of winning $1,000?
The latter. But, that all being said, people do not necessarily act rationally.
From: Eleutherophobics of the World...Unite!!!!!  Registered: Jul 2005
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M. Spector
rabblerouser
Babbler # 8273

posted 02 August 2008 05:03 PM
quote: Originally posted by Sven: ...and a transaction that neither would bother with because the net result is to (statistically) end up in the exact same place one started in prior to the transaction.
Just to make it clear: I understand the wager to mean that if the gambler loses, he loses $500, but if he wins he gets to keep the $500 wager plus an additional $500 in winnings.The original hypothesis was a onetime wager of $500 on a 50% chance of winning $1000. There's no way you could come out even; either you win $500 or you lose $500. Statistics only says you'll come out even in the long run, given an infinite number of iterations of the bet. Statistics is the science that tells us that if we lie down with our head in the oven and our feet in the refrigerator, on average, we will feel comfortable with the temperature. So when we say that statistically the house always wins, we know that in individual cases, the house doesn't always win. The "luck" fluctuates back and forth between the gambler and the house, and over time the house ends up "luckier". What actually motivates many gamblers is the hope that the natural variations in "luck" over time will swing their way, and they will come out ahead, knowing that other gamblers will come out behind in the natural order of things, in order to skew the "statistics" in favour of the house. So a gambler might bet a large sum of money on a single 5050 play in hope of a big return. Maybe she's desperate after a series of smaller losses. If she wins, she doubles her money and then goes home; that doesn't sound so unlikely to me. If you're feeling "lucky" why wouldn't a gambling person take the chance on a onetime bet? In fact, many make similar gambles at games like roulette, where they bet on either red or black, with a 5050 chance of winning, but a negative "expected value" regardless of how many times they repeat the wager, because the odds are in favour of the house. Similarly, many people make informal sports wagers with friends on a presumed "evenodds" basis. If someone would make a single wager of 50 cents on a 50% chance of winning a dollar, why wouldn't someone else be willing to wager $500 in the same circumstances to win $1000? The only difference is the extent to which the gambler is riskaverse and the amount of money he or she can afford to lose. As you yourself have noted, people will repeatedly play the slot machines when they actually face less of a chance of coming out ahead than they would with the "fair price" bet, if repeated over many iterations. 5050 odds on doubling your money is better than anything you will find at a casino, a lottery or a racetrack.
From: One millihelen: The amount of beauty required to launch one ship.  Registered: Feb 2005
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Martha (but not Stewart)
rabblerouser
Babbler # 12335

posted 04 August 2008 07:53 AM
quote: Originally posted by Sven: (h) Linda is a bank teller and is active in the feminist movement. (e) Linda is a bank teller. Reasoning:■ ... if she is a bank teller, I would guess she's more likely to be active in the feminist movement than notthus, (h) before (e).
The reason you offer would be a good reason for ranking (h) over (e*) Linda is a bank teller and is NOT active in the feminist movement. But the reason you offer is not a good reason for ranking (h) over (e). In fact, by the laws of probability, the probability of (h) is less than or equal to the probability of (e): in general, P(A&B) ≤ P(A). Here is a way of seeing this. Imagine a thousand people fitting Linda's profile, let's say a thousand Lindas. The number who satisfy (h) will certainly be ≤ the number who satisfy (e).
From: Toronto  Registered: Mar 2006
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M. Spector
rabblerouser
Babbler # 8273

posted 04 August 2008 05:14 PM
I think Sven's problem is in assuming that(e) Linda is a bank teller. means (e) Linda is a bank teller and is NOT active in the feminist movement. I explained in my [amended] first post why (h) is less likely than either (d) or (e). I notice remind also ignored my advice and put (c) ahead of (j), which is impossible. [ 04 August 2008: Message edited by: M. Spector ]
From: One millihelen: The amount of beauty required to launch one ship.  Registered: Feb 2005
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torontoprofessor
rabblerouser
Babbler # 14260

posted 05 August 2008 12:42 PM
[pedantry accelerates]A and B are independent if and only if (roughly) neither A nor B makes the other more or less probable. So A and B are dependent if and only if (roughly) either A increases the probability of B, or A decreases the probability of B, or B increases the probability of A, or B decreases the probability of A. It is plausible that being active in the feminist movement marginally decreases the probability of being a bank teller: maybe people active in the feminist movement are marginally less likely to work for a bank than people in the population at large. If this is so, then being active in the feminist movement and being a bank teller are not independent. Note that we have our conjunction inequalities regardless of whether or not they are independent.
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