Mckinsey Interview Questions | Biased Coin
Question
How can you tell if a given coin is biased?
(Hint- We want the answer in terms of Hypothesis Theorem)
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Statistics
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Answers ( 3 )
Just do a Chi square test for goodness of fit.
Ho – The coin is not biased (the probability of head is 0.5)
H1 – Coin is biased (the probability of head 0.5)
Assume we are allowed to toss the coin only 10 times, and on the basis of the outcomes, make our decision.
The decision rule is based on a critical value–if the number of heads is greater than or equal to the critical value, the null hypothesis is rejected–otherwise the null hypothesis is accepted.
null hypothesis is that the coin is fair (p=0.5) and the alternative hypothesis is that the coin is biased in favor of heads (p>0.5). Suppose the coin is tossed 10 times and 8 heads are observed. Since the alternative hypothesis is p>0.5, more extreme values are numbers of heads closer to 10. So, to compute the p-value in this situation, you need only compute the probability of 8 or more heads in 10 tosses assuming the coin is fair. But, the number of heads in 10 tosses of a coin assuming that the coin is fair has a binomial distribution with n=10 and p=0.5. The p-value is P[8 heads] + P[9 heads] + P[10 heads]. From the binomial probability distribution, P[8 heads]=0.044, P[9 heads]=0.01, and P[10 heads]=0.001. Thus the p-value is 0.044+0.010+0.001=0.055.
Now that the p-value is computed, how do you decide whether to accept or reject the null hypothesis? Since the p-value is simply the probability of getting the observed number of heads under the assumption that the null hypothesis is true, if this probability is small, it is unlikely that the null hypothesis is true. So ‘small’ p-values lead to rejection of the null hypothesis. But ‘small’ is not defined. The definition of small is up to the reader–if in the opinion of the reader, the p-value is small, the null hypothesis is rejected, while larger values would cause the null hypothesis to be accepted. In statistical practice, ‘small’ values are usually 0.10, 0.05, or 0.01. In the coin tosses above, the p-value is 0.055, and if a ‘small’ p-value for you is 0.05, you would fail to reject the null hypothesis, that is, you would say 8 heads in 10 tosses is not enough evidence to conclude that the coin is not fair.