Probability of flipping a coin

I know that the probability of flipping a coin 3 times and guessing the right answer is 0.125 but what happens to that rating of consecutive wins when the next flip is guessed wrong. Does it go back to .25 when guessing wrong once? Or do u need to separate the consecutive win ratio and the losses because they do not mix. Reason being is I flipped a coin and guessed it right 220 times out of 240 flips. Only 20 wrong guesses. I want to be able to rate how hard it was to pull of that experiment.

I am wondering because the only information I have been able to find online from searching about coin flips is that there is a consecutive probability like guessing right 5 times is .5 x .5 x .5 x .5 x .5 = .03125. Also I found little info about when flipping 100 coins u will always most likely get around 50/100 guesses right so the chances get slimmer the more accurate or inaccurate u get. I heard that getting 60/100 guesses right is about 0.0285 but I am not sure how they calculated it. I am pretty sure I am asking about two different probabilities. But I need facts.

Quote: Miracle0033

I know that the probability of flipping a coin 3 times and guessing the right answer is 0.125 but what happens to that rating of consecutive wins when the next flip is guessed wrong. Does it go back to .25 when guessing wrong once? Or do u need to separate the consecutive win ratio and the losses because they do not mix. Reason being is I flipped a coin and guessed it right 220 times out of 240 flips. Only 20 wrong guesses. I want to be able to rate how hard it was to pull of that experiment.

I am wondering because the only information I have been able to find online from searching about coin flips is that there is a consecutive probability like guessing right 5 times is .5 x .5 x .5 x .5 x .5 = .03125. Also I found little info about when flipping 100 coins u will always most likely get around 50/100 guesses right so the chances get slimmer the more accurate or inaccurate u get. I heard that getting 60/100 guesses right is about 0.0285 but I am not sure how they calculated it. I am pretty sure I am asking about two different probabilities. But I need facts.

The probability of an event with probability p happening exactly K times out of N is Combin(N,K) x p^N x (1-p)^K-N, where combin(N,K) is the number of combinations of N items taken K at a time.
For 60 heads out of 100 coin tosses, p = 1/2, N = 100, and K = 60; the probability of getting exactly 60 heads is about 0.0108.
The probability of getting 60 heads or more in 100 tosses is 0.02844.

However, going back to your original question, the probability of a coin coming up heads or tails does not change based on the past tosses. If I am calculating this right, the probability of correctly calling 220 or more coin tosses is the same as always guessing "heads" and having 240 tosses come up heads 220 times or more, which is 1 in 2 x 10⁴³. Of course, this also assumes that you are not cherry-picking a particular set of 240 consecutive tosses out of a much larger group.

Quote: Miracle0033

Reason being is I flipped a coin and guessed it right 220 times out of 240 flips. Only 20 wrong guesses.

I would say guessing a 50/50 event correctly 220 out of 240 times would be the most amazing feat in human history. It would happen 1 out of ..... ( too big a number for me to comprehend) times. Congratulations!

Quote: SOOPOO

I would say guessing a 50/50 event correctly 220 out of 240 times would be the most amazing feat in human history. It would happen 1 out of ..... ( too big a number for me to comprehend) times. Congratulations!

To 15 significant decimal places (curse you Excel!), the probability of 220 or more correct guesses out of 240 is 1 in 21,961,329,082,730,600,000,000,000,000,000,000,000,000,000.

To put that in perspective, that is like rolling a "yo" 34.5 times in a row.

Put another way, it would be like winning the Powerball jackpot 5 drawings in a row.

Quote: Wizard

To 15 significant decimal places (curse you Excel!), the probability of 220 or more correct guesses out of 240 is 1 in 21,961,329,082,730,600,000,000,000,000,000,000,000,000,000.

To put that in perspective, that is like rolling a "yo" 34.5 times in a row.

Well, if you want an exact answer, the probability is 40,226,323,692,033,636,822,768,162,719 / 2²³⁹

The exact expanded answer is 1 in :

883423532389192164791648750371459257913741948437809479060803100646309888 / 40226323692033636822768162719

=~1 in:

21,961,329,082,730,572,471,363,676,326,731,970,686,043,590.5442063594364368445588084638033680608852114041846136171781126370968084266066930535946323726147599859

You get this by taking the reciprocal of the integral from zero to infinity of:

(x/2)^219/219!*1/e^x*(1+(x/2)+(x/2)^2/2!+(x/2)^3/3!+(x/2)^4/4!+(x/2)^5/5!+(x/2)^6/6!+(x/2)^7/7!+(x/2)^8/8!+(x/2)^9/9!+(x/2)^10/10!+(x/2)^11/11!+(x/2)^12/12!+(x/2)^13/13!+(x/2)^14/14!+(x/2)^15/15!+(x/2)^16/16!+(x/2)^17/17!+(x/2)^18/18!+(x/2)^19/19!+(x/2)^20/20!)*1/2 dx