# Incredible coin toss story-what is its probability?

1. Jul 7, 2011

### Megaritz

Incredible coin toss story--what is its probability?

Today my grandfather told me a story, which he insists is true and which I believe he believes is true, which seems to defy all probability to an incredible degree.

He told me that many years ago, when he was taking a psychology class with Pat Norwood (who later became a professor) at Southwest Texas State Teacher's College (now called Texas State University), he was given a test without having studied for it. He knew none of the answers.

The test had 100 questions, each with 2 possible answers (True or False). The teacher explained that for guessing a correct answer, 1 point would be given, for a possible maximum of 100 points on the test. For every wrong answer, 2 points would be deducted, for a possible minimum score of -200 points. For every answer left blank, no points would be given or deducted.

My grandfather had a silver dollar, which he flipped for every question; he flipped it 100 times. For heads, he would mark "True," and for tails he would mark "False." He got a score of 86 points, meaning he got 93 answers right. All on his coin flips.

He tied for highest score with another student, who had been answering every question by flipping a nickel and also got 93 answers right.

My grandfather swears this story is true, and that after the graded tests were handed back by the astonished teacher, the other students in the class were amazed and carefully looked over the two coin-flippers' tests; they had been properly graded and both my grandfather and the other coin-flipper had each gotten 93 answers correct based on coin tosses, although the answers they got wrong were different for each of them.

I am not good with probability, but I tried to figure out how unlikely this scenario was to come about in this way. I figured the probability of this result was 1 out of 2^186, which is 2^93 * 2^93, which I assumed to be the probability of both my grandfather and the other student, individually, getting the scores they did via coin tosses; as far as I know, the probability of two events both happening is obtained by multiplying the probabilities of each individual event, separately.
And I based those numbers I got on the assumption that 93 coin tosses out of 100 leading to the correct answers (1/2 as the chance of a correct answer for each) was of equal probability to 93 coin tosses out of 100 being heads --now disregarding the fact that "heads" originally corresponded to "true" which only around half the time was "correct", by substituting "correct" for "heads" just for ease of thinking clearly about the coin. My result also involved the questionable assumption I made that 93 coin-toss results of heads out of 100 is no more or less likely than 93 consecutive results of heads, which has a probability of 2^93. I apologize if my writing of my reasoning here is unclear, and I will try to clarify if needed, although anyone else's math is likely to be better than mine anyway.

2^186 is a 55 digit number (or 56, or something--shows how well I know calculator notation), which is enough to make me question either the math/reasoning I did to find it, or suspect something is up with the story itself. As I said, I don't know much about probability, so I'd appreciate it if someone could confirm how improbable Grandpa's story is. At every assumption I make in my calculations, I wonder whether I reasoned erroneously. If someone could tell me what the probability is of two people both getting 93 correct answers on a True-False test using coin tosses, I would very much appreciate it. Thank you.

Last edited: Jul 7, 2011
2. Jul 8, 2011

### willem2

Re: Incredible coin toss story--what is its probability?

If you get 93 questions right and 7 wrong, your score will be

+1*93 - 2 * 7 = 79.

A score of 86 isn't possible when answering all questions.

The probability to get k out of n questions right is

$$2^{-n} \frac { n!} {k! (n-k)! }$$

This works out to 1.26*10^(-20) for a single test.

3. Jul 8, 2011

### pmsrw3

Re: Incredible coin toss story--what is its probability?

First, there's a problem with the story as told. If scoring is +1 for right answers, -2 for wrong, and your grandfather answered every question and got 93 right, his score would have been 93 - 14 = 79, not 86. It is in fact impossible to answer all questions and get 86 with this scoring system. I assume the score for incorrect answers was actually -1.

OK. It's very improbable, but not quite as improbable as you calculated. In answering a question of this type, you shouldn't calculate the probability of getting the exact result that you got, because that's almost always a very small number, even for unremarkable events. For instance, suppose there were 30 students in the class, and they all answered by flipping coins. The most probable result is that every student got 50 questions right and 50 wrong. But the probability of this most probable result is 10^-33!

So instead of "What is the probability of the event that happened?", you need to ask "What is the probability that something this far from the expected result could happen?" That means, first of all, that you have to add up all the different ways your grandfather could have gotten 93 questions right. There are $\frac {100!} {93! 7! } = 16,007,560,800$ of those. Multiply that by the chance of any particular set of 100 answers, which is 1/2^100, and you get the formula willem2 gave. But you're not done yet, because you also have to consider that you grandfather would have been telling you this story if he got 94 right, 95, right, etc. Under that principle that we include everything as unexpected as what happened, we have to add those in, too. That turns out not to be a big correction: it brings up the probability to 1.36 x 10^-20. Finally, we have to ask for the probability that this happened to 2 or more students in the class. To calculate that I really need to know the class size, but if I guess 30, it comes to 8 x 10^-38.

It's still a ridiculously small number, and it's hard to believe that it occurred exactly as your grandfather described (especially since there's one detail of the story that's obviously wrong). But it's not quite as ridiculously small as 2^-186.

Last edited: Jul 8, 2011
4. Jul 8, 2011

### Megaritz

Re: Incredible coin toss story--what is its probability?

I was the one who figured, incorrectly, that a score of 86 meant 93 correct answers. I hadn't thought it out carefully enough to realize the score with 93 correct answers would be 93 - 14 instead of 100 - 14. I told Grandpa, and he said he definitely scored in the high 80s. If it was a score of 88, that meant he got 96 right--assuming I haven't made another mistake. He reassured me that 2 points were taken off for each incorrect answer, since there were students who got scores of less than -100.

I've been trying to think of scenarios in which the probability wouldn't be so low as it seems. I suggested the teacher could have made up the results to mess with them, but apparently he gave the tests back to the students, who then looked over the coin-flippers' tests, so assuming his memory about that is correct, no foul play was possible on the teacher's part.

I also wonder if the other student had actually been flipping his coin just to pretend to use it as a means of answering questions; he was sitting in the back of the class and could have seen Grandpa flipping his coin, so maybe he realized Grandpa was flipping his coin to get his answers, and thought it would be funny to act like he was doing the same thing. If so, then the other student had actually gotten the score he did because he'd studied for the test, and so the only coincidence as far as he's concerned was that he also got 96 answers right. If that were the case, only one person would have gotten 96 answers right by coin tosses.

In addition, I wonder if Grandpa remembers the number of questions incorrectly, and there had actually been fewer than 100. I think the probability of the story being completely correct is so small that any other explanation is less far-fetched.

5. Jul 8, 2011

### I like Serena

Re: Incredible coin toss story--what is its probability?

Here's an alternative scenario.

The other 14 he was about 67% sure, so he flipped a coin, whether he'd answer or not.
5 he didn't answer, scoring no points.
From the 9 he answered he had 6 right and 3 wrong, scoring effectively 0 points.
So he totalled 86 points.

Of course, with every passing year the story became more impressive.

6. Jul 8, 2011

### pmsrw3

Re: Incredible coin toss story--what is its probability?

What I was thinking (similar to ILS) is that your grandfather likely had a pretty good idea for most of the questions, and flipped a coin for the rest. In his memory it has turned into flipping a coin for all the questions.

Memory is much more fallible than people generally realize. I'm 55, and I have distinct memories of all kinds of things that never happened. ("When I was younger, I could remember anything, whether it had happened or not; but my faculties are decaying now and soon I shall be so I cannot remember any but the things that never happened." -- Mark Twain's Autobiography)

7. Jul 8, 2011

### Megaritz

Re: Incredible coin toss story--what is its probability?

He insists he had not even looked at the chapter the test was on, and that he flipped a coin for every single question, and that the teacher had watched him flipping the coin from the beginning of the test and had told him the odds of him getting the score he did were astronomical. Of course, it was quite a few years ago and there are any number of things he could have remembered wrongly, but I still wonder exactly how it really happened.