(adsbygoogle = window.adsbygoogle || []).push({}); Incredible coin toss story--what is its probability?

Today my grandfather told me a story, which he insists is true and which I believehebelieves 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 anickelandalsogot 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.

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# Incredible coin toss story-what is its probability?

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