Probability Question

[JohnGalt2341]

If I were to glue together a Dime and a Penny and I wanted to know what percentage of the time it would land Penny side up versus Dime side up when flipped... what would be the MINIMUM number of times I would have to flip it before I would have an accurate percentage?

[FourLengthsClear]

In this instance, accurate is a relative term.

What confidence level would you want/need your results to be?
Would you be happy, for example with results which said that it should land dime side up between 60 and 62 percent of the time?

[JohnGalt2341]

In this instance, accurate is a relative term. What confidence level would you want/need your results to be? Would you be happy, for example with results which said that it should land dime side up between 60 and 62 percent of the time?

Within 3% would be good.

[MonkeyF0cker]

http://en.wikipedia.org/wiki/Checking_whether_a_coin_is_fair

[JohnGalt2341]

http://en.wikipedia.org/wiki/Checkin...a_coin_is_fair

Thanks for the link although much of that was over my head. My coin(half dime, half penny) is definitely not a Fair Coin which is why I ask the question. I realize after a million flips the percentage would be incredibly accurate. But how accurate would it be after say.... 500 flips?

[bztips]

In addition to how accurate you want to be in terms of +/- pct, you also have to decide how confident you want to be in that interval. The formula turns out to be very simple for an approximate 95% confidence interval (equating to 2 std deviations around your estimate):

Assuming the true percentage is not too close to 0 or 1, the required number of trials is 1 / (interval width/2)^2 (hope my math is right!)

In other words, if your interval width is .06 (which corresponds to +/- .03), then you need 1 / .03^2 trials to be 95% confident that the percentage is within +/- 3% of your observed sample mean. This works out to 1,111.

If you need better accuracy than that, say 99% confidence -- which corresponds to 3 std deviations -- then you need to multiply further by a factor of 1.5^2.
1111 * 1.5^2 = 2500, meaning you need 2500 flips to be 99% confident that the percentage is within +/- 3%.

On the other hand, if you were willing to accept, say +/-5% with 95% confidence, then you need only 1 / .05^2 = 400 flips.

[CanuckG]

P(k) = nCk * p^k * (1-p)^(n-k) = 3%

where p = probability of Penny side up, which you can't really determine without knowing how the penny/dime is weighted.

Solve for k

I take no credit for this.

[mjespoz]

n=p*(1-p)*[(z/d)^2]
where p=probability of event (if unknown then use p=0.5, this maximises n for conservativeness), z=critical value for the chosen confidence level (e.g. for 95% then z=1.96) & d=the precision (e.g. 3% or 0.03)
n=0.5*0.5*(1.96/0.03)^2
n=1067.11 = 1068 (rounding up).

Hope that this helps.

Cheers,
mjespoz

[JohnGalt2341]

Thanks a bunch guys! I think I'm going to run the test myself within the next week or so and see how much the percentages vary per 100 flips. I'll post the results here if by chance anyone is interested in the outcome.

[JohnGalt2341]

Results after first 100 flips: Dime side up=57. Penny side up=43. I'm not too surprised so far.

[cyberbabble]

coin flip bias

Some interesting points for coin flippers/spinners. Maybe flipping a coin is not really 50/50.

http://www.codingthewheel.com/archives/the-coin-flip-a-fundamentally-unfair-proposition

[JohnGalt2341]

Some interesting points for coin flippers/spinners. Maybe flipping a coin is not really 50/50. http://www.codingthewheel.com/archiv...ir-proposition

Thanks for the link. That's very interesting.

Here are the results to my 2nd set of 100 flips:
Penny Side Up=53
Dime Side Up=47
Overall so far:
Dime Side Up=104/52%
Penny Side Up=96/48%

[Boxing Champ]

What are you trying to achieve?

[JohnGalt2341]

What are you trying to achieve?

A couple of things. #1 I want to see how many flips it takes before the percentages start to level off. As far as I know nobody has ever glued together a Dime and a Penny before and done this experiment. Therefore the percentages of it landing Penny side up vs Dime side up are virtually unknown. #2 In my mind the coin landing Dime side up is supposed to represent a very good Sports Handicapper. I'm going to predict that when the percentages do level off it will be somewhere around 56% Dime side up and 44% Penny side up. I could be way off of course. If a professional Handicapper could consistently win 56% of his bets at -110 he could become a very rich man. As you can see after 200 flips Dime side up is only hitting 52%. This would be a small deficit if making 200 bets at -110. This however doesn't necessarily mean that it is going to stay at 52%. In gambling there are hot streaks and cold streaks just as there are streaks when flipping a coin, especially a coin that is half Dime/half Penny. Ultimately I want to see the extremes on both ends and find out how many flips/bets it takes until you know the true percentages of both a coin flip and a handicapper.

[JohnGalt2341]

Thanks for the link. That's very interesting. Here are the results to my 2nd set of 100 flips: Penny Side Up=53 Dime Side Up=47 Overall so far: Dime Side Up=104/52% Penny Side Up=96/48%

3rd set of 100 flips results:
Penny Side Up=51
Dime Side Up=49
Overall so far:
Dime Side Up=153/51%
Penny Side Up=147/49%

I must say that I'm pretty surprised that the results are so even thus far.

[JohnGalt2341]

3rd set of 100 flips results: Penny Side Up=51 Dime Side Up=49 Overall so far: Dime Side Up=153/51% Penny Side Up=147/49% I must say that I'm pretty surprised that the results are so even thus far.

Results after 4th set of 100 flips:
Penny side up=52
Dime side up=48

Overall so far:
Dime side up=201/50.25%
Penny side up=199/49.75%

Amazingly, the Penny being glued to the Dime seems to have almost no effect on the outcome so far. Perhaps I should have glued together a Quarter and Nickel.