Coin Flip Probability

Does anyone have a chart (or any other resource) that lists the probability of a coin flip landing on heads (or tails) 1 out of 1 trials, 2 out of 2, 3 out of 3, ect. to possibly 20 out of 20?

You mean the powers of 1/2 ?

1 0.5000000
2 0.2500000
3 0.1250000
4 0.0625000
5 0.0312500
6 0.0156250
7 0.0078125
8 0.0039063
9 0.0019531
10 0.0009766
11 0.0004883
12 0.0002441
13 0.0001221
14 0.0000610
15 0.0000305
16 0.0000153
17 0.0000076
18 0.0000038
19 0.0000019
20 0.0000010

Coin flips are not 50/50 propositions. Stanford University professor, Persi Diaconis, has demonstrated that a coin will land with the same pre-flip face up 51% of the time.

While I have heard this claim, I am still highly skeptical of that conclusion. They initially considered a high-precision flipper that gave repeatable results, and if you know anyone who can flip coins (hand-held, flipped by the thumb, higher than the head) that repeatedly, then you have probably met Lieutenant Commander Data. I really doubt that their attempts to analyze the motion of flipping coins from video tapes could ever be really conclusive of something that could be used to advantageously wager on a coin flip. Even the reference that you cited just says, "Preliminary analysis of the video-taped tosses suggests that a coin will land the same way it started about 51 percent of the time." Note that "preliminary", "suggests", and "about" leave this far from being a well-established conclusion.

Quote: rebbit

Does anyone have a chart (or any other resource) that lists the probability of a coin flip landing on heads (or tails) 1 out of 1 trials, 2 out of 2, 3 out of 3, ect. to possibly 20 out of 20?

Your question is not clear. As the previous post responded it sounds like you are asking for 1/2^j for j = 1...20

I will try and answer the question I think you are asking. You are asking about the specific case of a casino and player having even odds with no house advantage. You want to know the odds of a player losing 2 in a row out of only 2 tosses, then out of 3, 4, 5, ... up to 20.

The table gives the answer for up losses from 1 in a row, up to 9 in a row, out of 1 to 20 games. I could do it for streaks up to 20, but the probabilities start getting very small.

The first column is the simple case of the probability of getting a simple loss (i.e. the probability that the casino does not win every game).

Is this table what you are looking for?

casino wins (p)	50.00%
player wins (q)	50.00%
Pkk=p^k	50.00%	25.00%	12.50%	6.25%	3.13%	1.56%	0.78%	0.39%	0.20%
DELTA=q*Pkk	25.00%	12.50%	6.25%	3.13%	1.56%	0.78%	0.39%	0.20%	0.10%
Average	2	6	14	30	62	126	254	510	1022
Losses in a row	a loss	two in a row	three	four	five	six	seven	eight	nine
1	50.0000%
2	75.0000%	25.00%
3	87.5000%	37.50%	12.50%
4	93.7500%	50.00%	18.75%	6.25%
5	96.8750%	59.38%	25.00%	9.38%	3.13%
6	98.4375%	67.19%	31.25%	12.50%	4.69%	1.56%
7	99.2188%	73.44%	36.72%	15.63%	6.25%	2.34%	0.78%
8	99.6094%	78.52%	41.80%	18.75%	7.81%	3.13%	1.17%	0.39%
9	99.8047%	82.62%	46.48%	21.68%	9.38%	3.91%	1.56%	0.59%	0.20%
10	99.9023%	85.94%	50.78%	24.51%	10.94%	4.69%	1.95%	0.78%	0.29%
11	99.9512%	88.62%	54.74%	27.25%	12.45%	5.47%	2.34%	0.98%	0.39%
12	99.9756%	90.80%	58.37%	29.88%	13.94%	6.25%	2.73%	1.17%	0.49%
13	99.9878%	92.55%	61.72%	32.42%	15.41%	7.02%	3.13%	1.37%	0.59%
14	99.9939%	93.98%	64.79%	34.87%	16.85%	7.78%	3.52%	1.56%	0.68%
15	99.9969%	95.13%	67.62%	37.23%	18.26%	8.54%	3.90%	1.76%	0.78%
16	99.9985%	96.06%	70.23%	39.50%	19.65%	9.29%	4.29%	1.95%	0.88%
17	99.9992%	96.81%	72.62%	41.69%	21.02%	10.03%	4.67%	2.15%	0.98%
18	99.9996%	97.42%	74.82%	43.80%	22.37%	10.77%	5.06%	2.34%	1.07%
19	99.9998%	97.91%	76.84%	45.84%	23.69%	11.51%	5.44%	2.54%	1.17%
20	99.9999%	98.31%	78.70%	47.80%	24.99%	12.23%	5.82%	2.73%	1.27%

One of the rows is the average number of games you expect to play to get a loss of 1,2,3, ... 9 in a row .

Yes that is exactly what I was looking for! Thank you very much for your effort, I really appreciate it.

Quote: rebbit

Yes that is exactly what I was looking for! Thank you very much for your effort, I really appreciate it.

As long as you understand the table. For instance if you are interested in the second column there is a 25% chance of losing two in a row if you toss the coin 2 times, and there is a 50% chance of losing two or more in a row if you toss the coin 4 times (but that includes cases where you have lost the first 2, the first 3, etc.)

The other way to look at the problem is if you toss the coin AND STOP after you have two losses in a row. Sometimes that will happen in only 2 coin tosses, and sometimes you will toss it 30 or more times. But once you have two losses in a row, you write down the number of coin tosses to get to that point. If you repeat this experiment a large number of times, the average of the required number of coin tosses will be 6.

The sequence that represents the average is very easy to generate. Start with 2, then add 1, then double.
2,(2+1)*2=6,(6+1)*2=14, ...

So your expected number of coin tosses before you get 20 "losses" in a row is slightly over 2 million (where the player designates either a head or a tail as a loss)
1 : 2
2 : 6
3 : 14
4 : 30
5 : 62
6 : 126
7 : 254
8 : 510
9 : 1,022
10 : 2,046
11 : 4,094
12 : 8,190
13 : 16,382
14 : 32,766
15 : 65,534
16 : 131,070
17 : 262,142
18 : 524,286
19 : 1,048,574
20 : 2,097,150


I am very specifically using the term loss where in the case of coin tossing you decide of a head or a tail is a loss. There is a related question where you are simply looking for streaks of an arbitrary length and you don't care if they are a streak of heads or tails.

Since many people like to gamble with a "double or nothing" strategy (otherwise known as a Markov strategy), they usually have a bankroll that allows them to double 5 or 6 times. In most casino games the house minimum and maximum will only allow you to double 6 times before you go over the maximum. The largest spread I have ever seen in a casino let you double 9 times. There is a mistaken belief by many people that the strategy is more effective if you can find such a table.

-

It exist other ways to apply math and probability using coin tossing.
One way can be to use the law of series - is not really a law and you can call it observations of the distribution.
The bell curve has no limits - but still you can measuring balance and imbalance.

Lets say singles outcomes are one side of the coin and series are the other side of the coin.
Then you can apply math and probability and notice that 3 std is not so rare.

Series has the value of 1
Singles has the value of 1

Then if you get 14 series to chop with only two singles present - you would have 3 std.
The waves of balance and imbalance comes in different states - there exist three states.

Calculation

First you have to get the Absolute Std when you calculate.
Then you take 14 - 2 = 12

Now we want to get the statistical std so we continue with...
14 + 2 = 16

Now we take the sqr of 16 = 4

And finally we divide the Absolute Std whit the sqr

12 sqr 4 = 3,00

The Statistical STD 3,00

Dummy chart:
Don't know how to add img tag ...



It exist many other ways to divide the law of series and the distribution.

Quote: Ayecarumba

Stanford University professor, Persi Diaconis, has demonstrated that a coin will land with the same pre-flip face up 51% of the time.

Don't forget that Persi Diaconis used to be a magician. He could draw on his skills to demonstrate that you have two left feet.

John Scarne also used to be a magician. Is a magician someone you can trust?

Study says coin lands on the side it started on 50.8% of the time. I dunno, is 350,757 flips a large enough sample size?

https://www.engadget.com/coin-flips-dont-appear-to-have-5050-odds-after-all-171556415.html?guccounter=1

Quote: MichaelBluejay

Study says face-up coin lands face-up 50.8% of the time. I dunno, is 350,757 flips a large enough sample size?

https://www.engadget.com/coin-flips-dont-appear-to-have-5050-odds-after-all-171556415.html?guccounter=1
link to original post

My natural coin flip only goes 6 inches high when sitting. When standing, I tend to go 8-12 inches. I guess it flips in the air no more than 15 times. I have no problem believing that this is not enough to randomize the result to within 1%.

An NFL coin flip starts four feet off the ground and has much more time to flip in the air than a coin flip that is caught in hand.

Also note that the total number of heads and tails will be very close to 50-50 if the coin is not deliberately reset to a heads or tails position between each flip. If you start with heads up, even a bad flipper will start the second flip from a fairly random starting point. By the 10th flip, the results will be essentially random. I think the NFL should have the ref flip the coin into his hand and then start the real flip with that side up just to be fair.

Quote: MichaelBluejay

Study says face-up coin lands face-up 50.8% of the time. I dunno, is 350,757 flips a large enough sample size?
link to original post

Just for clarity, it's saying the coin will land on the side that started face up 50.8% of the time. I could see some interpreting "face up" to mean "heads."

One standard deviation is 296 or 0.0844%. The probability of being off 0.8% with a fair coin with a 350,757 sample size is 1 in 756,651,780,485,024,000,000.

The thing that bothers me about the study is that they aren't reporting a subset of people who have the opposite bias. Let me assume that most people flip the coin fairly reproducibly so that it flips N/2 times on average with a fairly narrow distribution. Let us say N = 10, then the coin is most likely to rotate 5 full times before it is caught. This would produce the observed bias. However, if N is odd, then there should be the reverse bias.

I could believe that the most common flipping techniques have N coming out even, but I see no reason why there would be a dearth of flippers who have have a distribution centered on an odd N. The size of the coin should affect N dramatically. The higher N is, the less bias should be observed.

If the coin is not flipped so that it rotates on a pure horizontal axis, but also has some spin, then gyroscopic effects will introduce some bias. I am flipping the coin off of my thumb and index finger. This gives a higher rate of rotation than simply tossing the coin off the end of the fingers. It might introduce off-axis spin, but my flips come out looking like the coin is flipping around a horizontal axis.