Question 8.1.11: We have two coins: one is a fair coin and the other is a coi......
We have two coins: one is a fair coin and the other is a coin that produces heads with probability 3/4. One of the two coins is picked at random, and this coin is tossed n times. Let Sn be the number of heads that turns up in these n tosses. Does the Law of Large Numbers allow us to predict the proportion of heads that will turn up in the long run? After we have observed a large number of tosses, can we tell which coin was chosen? How many tosses suffice to make us 95 percent sure?
No, we cannot predict the proportion of heads that should turn up in the long run, since this will depend upon which of the two coins we pick. If you have observed a large number of trials then, by the Law of Large Numbers, the proportion of heads should be near the probability for the coin that you chose. Thus, in the long run, you will be able to tell which coin you have from the proportion of heads in your observations. To be 95 percent sure, if the proportion of heads is less than .625,predict p = 1/2; if it is greater than .625, predict p = 3/4. Then you will get the correct coin if the proportion of heads does not deviate from the probability of heads by more than .125. By Exercise 7, the probability of a deviation of this much is less than or equal to 1/(4n(.125)2). This will be less than or equal to .05 if n > 320. Thus with 321 tosses we can be 95 percent sure which coin we have.