The laws of probability, so true in general, so fallacious
in particular.
—Edward Gibbon
This is the first
of a short series on probability and statistics. You remember, the bit about if you have 3
white marbles and 3 black marbles and you simultaneously draw 4 random marbles
what are the odds of grabbing exactly 2 black ones? (Answer: 9/15 or 60%).
We are bombarded by probabilities and statistics everyday,
yet for most of us our intuition actually works against us. Like the guy who thinks that the more lottery
tickets he buys, the better his chance of winning money. In
fact, the more lottery tickets you buy, the better your chance of losing money.
Think of it this way: If you bought one lottery ticket, you might
hit the jackpot. But if you bought all the lottery tickets, then (since the
total winnings are always less than the total income generated from ticket
sales) you are guaranteed to lose. Put
another way, buying more lottery tickets for a specific draw will increase your
chance of a winning ticket, but will decrease your chance of winning
money. This is because the odds are
stacked against you from the start (the "house" always ends up
ahead), and therefore every dollar you spend, in a probabilistic sense, loses
you money. As a poker pro would tell you, the entire scheme has a "negative expectation."
In other words, gambling is a tax for people who can't do
math.
This is especially true for the case of BC, where the
government runs the lottery and, being the government, affords itself shameless odds. Take the game Sports
Action. All you need to do in this
gamble is to call the winner three sports games correctly and you're a winner.
If you're a die-hard hockey fan, what could be easier? Just pick three sure things and you're off
to the bank. Well, not really. What's a "sure thing" in hockey? Say Vancouver versus Edmonton in 2011 (or
Oilers versus Canucks in 1986, if you prefer).
Even then your odds are maybe 80%.
So what are the odds of calling two sure things? That would be 80% x 80% which is (0.8)2
= 0.64. Two sure things and we're
already down to a 64% chance. Calling
three games would be (0.8)3 = 51.2%.
So intuitively you might think you have an 80% chance of calling three
"sure things" but in reality your odds are little better than
50:50. Did I mention that if they do a
lousy job on the odds and the lotery company that runs Sports Action
ends up losing money in a given weeks, they can cancel the whole week's tickets and not
pay anyone out? That, my friends, is
what is known in the business as a sucker bet.
OK, try this one:
You’ve
flipped Heads 9 times in a row. What are
the odds of flipping Heads again?
Got your answer?
Here’s where things get tricky. The odds of flipping Heads on a fair coin ten
times in a row are easy to figure out: 1
in 210 or 1 in 1,024. So it
might be intuitive to you that the odds of you flipping one more Heads is 1 in
1,024. I mean,
there's no way you're going to keep that streak going! But this fails to account for the fact that
you’ve already flipped nine Heads in
a row. The odds of that were (before you
started) 1 in 29 or 1 in 512.
But now that you’ve already done it, the probability is 1. A probability of 1 means 100% or
certainty. And you are certain that
you’ve flipped nine Heads in a row. So
thinking about it, even though you’ve already flipped nine Heads in a row, your
odds of flipping Heads again is simply 1 in 2, otherwise known as 50:50. Probabilities predict the unknown or the future;
the known past has a probability of 1 because it happened.
Want another mind-bender?
How about this:
Say you plan to roll a die 20 times. Which
of these results is more likely: (a) 11111111111111111111, or (b)
66234441536125563152?
Marilyn vos Savant, who was famous for having the Guinness Book
of World Record highest IQ in the 80s (before, apparently, they did away with the highest IQ record), answered:
In theory, the results are equally likely.
Both specify the number that must appear each time the die is rolled. (For
example, the 10th number in the first series must be a 1. The 10th number in
the second series must be a 3.) Each number—1 through 6—has the same chance of
landing faceup.
But let’s say you tossed a die out of my view and then said that the results were one of the above. Which series is more likely to be the one you threw? Because the roll has already occurred, the answer is (b). It’s far more likely that the roll produced a mixed bunch of numbers than a series of 1’s.
But let’s say you tossed a die out of my view and then said that the results were one of the above. Which series is more likely to be the one you threw? Because the roll has already occurred, the answer is (b). It’s far more likely that the roll produced a mixed bunch of numbers than a series of 1’s.
Is she right or wrong?
Think about that. I'd be interested in your comments.
Statistics and probability are important factors in a lot of
research science. But they are notoriously
difficult to understand, as the above puzzles demonstrate. As a matter of fact, of Ms. vos Savant's four
most controversial puzzles, all of them have been concerned with probabilities.
Let's hear your answers!
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth is not certain. The proposition of interest is usually of the form "Will a specific event occur.
ReplyDeleteIt is a tough one but I'm inclined to disagree with her. It's only human reasoning to think that there is no way you would actually get 20 ones in a row. Statistically, since the chance is equal for each number at each roll, the chances would be the same.
ReplyDeleteHave you ever meet anyone who choose the 6-49 numbers 1,2,3,4,5,6? Our logic would dictate the chances are less than whatever random numbers we may pick... But are they?
I'd say it's far more likely that the roll produced a bunch of fixed numbers - but equally likely that you roll a) or the exact sequence of b). In fact, in real life, I would say your odds of rolling a) are greater that b), as you get the sum of the probabilities that you rolled a) naturally, plus the probability that you have a fixed die.
ReplyDeletePut another way, the probability of the person with the highest IQ in the world being incorrect on a math problem rather than me is extremely high. But I've bet mentally with Stephen Hawking and one, so I'll stand with my answer without spending more time when I should be working.
Your analysis of gambling is only a mathematical one, not an economic one. For a gambler, the perceived marginal utility of winning a million dollars is much greater than the dollar spent on any given ticket.
Also, considering the following case: There is a yearly lottery where you pay $100,000 to buy a ticket which gives you a 1 in a 1000 chance to win 10 Billion dollars. You can expect to $100 for every dollar spent on tickets! For most of us, the negative value of losing $100,000 99.9% of the time is far greater than the value of being a multi-billionaire 0.1% of the time.
I think Ms Vos Savant's analysis is correct, but she is somewhat loose in the language of her response. Indeed the probability of rolling any specific sequence of numbers is the same given fair dice.
ReplyDeleteShe also states that you are far more likely to roll a set of mixed numbers than a series of 1's. This is true as far as it goes, but in this case we're not talking about a set of mixed numbers; we're talking about a specific sequence (66234441536125563152).
You would definitely be more likely to believe the second sequence, even though the odds are the same. If you rolled that many 1's, most people would be inclined to believe the die was suspect.
And despite the fact that both sequences are equally likely,
You would be correct to suspect something was going on - the odds of the dice being biased being astronomically* higher than the odds of rolling the sequence. If this weren't true, we would not be able to determine any P values in science.
ReplyDeleteInterestingly, to me, is that thinking about your blog led me in a few steps to this real life probability problem reminiscent of your post:
https://dl.dropbox.com/u/33382946/Tim%20Harford%20The%20Undercover%20Economist%20Chapter%206.pdf
*greater than I feel justified in calculating while I should be working.
Hi Adam,
ReplyDeleteYou were once an EUS president at UBC, correct? I am doing a story on engineering pranks, and I would love to speak with you. Could you contact me at gordon.katic [a] ubc . ca. ?