Heads occurs 15 times out of 20 - Is it a fair coin?

[phono]

I have to answer this question by using references to expectation and probability as well as any other relevant considerations. My textbook says that 96% of the time, the number of heads in 20 flips will be between 6 and 14 inclusive. Prove this.

Ok I can prove the answer in the textbook using Binomial Theorem.

Probability of k heads from n flips of a fair coin is n choose k = n!/((n - k)! k!) divided by total possibiliries = 2^n

Sp P(5) heads = (20 choose 5)/1048576

Sum P(n) from n = 6 to 14
(38760 + 77520 + 125970 + 167960 + 184756 + 167960 + 125970 + 77520 + 38760)/1048576

= 1005176/1048576 ~ 0.9586 ~ 96%

Edit: The question actually says "do you think the coin is fair?", sorry. Using references to expectation as well.

Thanks

 

[HallsofIvy]

You can't prove it is a fair coin. A fair coin can produce 15 heads out of 20 tosses and so can an "unfair" coin. What do you mean by an "unfair" coin? Perhaps if the probability of getting a given result with a fair coin were very low, but that result did come up you could say that the coin was unfair. But you need to decide what the cut off is: 10%? 5%? 1%? Then calculate the probability with a fair coin and see if it is below that.

Note that even a fair coin can produce 20 heads out of 20- just with very low proabability.

 

[astrosona]

OK, all the probability theorem and all the formula works for a large number of tries, you have to consider that you never can use probability formulas for a small number of tries (although we do this in problems! but mathematically and physically it is NOT correct!).

Probability of 1/2 for coins can be seen when you try it i.e. 100000 times not 20 times! this is the key point.

So you can never judge if the coin is fair or unfair in 20 times.

PS. unfair means if the coins is made correct and it has the same physics and shape and weight and ... in all it's place and one side is not heavier than the other side.

Good luck

 

[turbo]

Be aware that with consistent reflexes and timing in the coin-capture you can learn to influence the outcome of what looks like a casual coin-toss. Not relevant to the mathematical aspects of the question - just a heads-up about coin tosses in the real world.

Agreed, you can get relatively long strings of heads or tails even with an honest coin and random launches/catches, but increasing the sample size evens those out.