Question

What’s the probability that a fair coin results in 12 or fewer heads from 40 flips? And What does that probability say about the fairness of the coin?

Answer 1

Let X denotes the number of heads .

Given that the coin is fair so, P(X)= 0.5 and n=40

Here, X follows Binomial( 40,0.5)

So, P(X <= 12 ) = P(X= 1) + P( X= 2) + P(X = 3) +.....+P( X= 12)

= 0.008295

This can be calculated using Excel by = BINOM.DIST(12,40,0.5,TRUE) function

Nothing can be inferred about the fairness of coin by just looking at the cumulative probability.

First of all, you must understand that there is no such thing as a perfectly fair coin, because there is nothing in the real world that conforms perfectly to some theoretical model. So a useful definition of "fair coin" is one, that for practical purposes behaves like fair. In other words, no human flipping it for even a very long time, would be able to tell the difference. That means, one can assume, that the probability of heads or tails on that coin, is 1/21/2.

Whether your particular coin is fair (according to above definition) or not, cannot be assigned a "probability". Instead, statistical methods must be used.

Here, you make a so called "null-hypothesis": "the coin is fair". You then proceed to calculate the probability of the event you observed (to be precise: the event, or something at least as "strange"), assuming the null-hypothesis were true.Set your confidence interval say 95%

Now calculate p-value if p-valye < 0.05 then we may assume that our null hypothesis is rejected and hence coin is unfair

Otherwise Coin is fair