Ch. 5 SAMPLING FROM BINOMIAL POPULATIONS 115 



certain rules for acting upon the basis of sampling evidence. It 

 follows that the probability of making a correct decision from any- 

 specified future sample (say the next one we are going to take) also 

 can be stated. 



To illustrate some of the preceding discussion, suppose you are 

 about to engage in a coin-tossing game in which "heads" is the 

 event which is of particular interest to you. Assume, also, that you 

 are not satisfied that the coin is unbiased but are not going to woriy 

 about bias unless the probability of heads is as low as 1/3. Before 

 playing the game you are going to flip the coin 15 times and then 

 come to a decision regarding the bias of the coin. What rules for 

 action should you adopt and how effective will they be in detecting 

 bias as bad as p = 1/3? It is being assumed that you are not enter- 

 taining the possibility of bias toward too many heads. 



As long as the coin has two sides and one is heads, the other tails, 

 any result from to all 15 heads can occur on 15 flips regardless of 

 bias in the coin. However, it should be clear that the relative fre- 

 quencies of occurrence of the 16 possible results are dependent upon 

 the size of p. For p = 1/3, for example, such a result as 15 heads 

 on 15 throws is an extremely rare occurrence. The actual rarity, 

 in terms of probability, can be derived from the binomial series for 

 (q -j- p) n , with p, q, and n given. 



If p = 1/2 and n = 15, the binomial series is 



(1/2 + 1/2) 15 = .000 + .000 + .003 + .014 + .042 + .092 + .153 

 r: 1 2 3 4 5 6 



+ .196 + .196 + .153 + .092 + .042 + .014 + .003 

 r: 7 8 9 10 11 12 13 



+ .000 + .000. 

 r: 14 15 



When p is unknown and a sample has produced r = 0, 1, 2, or 3 heads 

 on 15 random flips, you probably would be very reluctant to accept 

 the hypothesis, H (p = 1/2) because the total probability of the 

 occurrence of one of these 4 mutually exclusive events is but .017, 

 or about 1 chance in 59. Although it is true that one of those 4 

 results can be obtained when the coin is unbiased — and you knew 

 this before you tossed the coin 15 times — you are now faced with 

 the necessity to decide if the coin is biased or not, and you must do 



