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THE POPULAR SCIENCE MONTHLY. 



CORRESPONDENCE. 



THE LOGIC OP PROBABILITIES. 



To the Editor of the Popular Science Monthly. 



IN the April number " Illustrations of 

 the Logic of Science," p. 706 the 

 writer says : " What is the probability of 

 throwing a six with one die? The antece- 

 dent here is the event of throwing a die ; 

 the consequent, its turning up a six. As the 

 die has six sides, all of which are turned 

 up with equal frequency, the probability 

 of turning up any one is one-sixth." Ad- 

 mitted ; but is not this also true : that if 

 you throw a single die, say twenty times, 

 and fail to turn up a six, the probability 

 thereafter of turning up a six is increased ? 

 One would say so ; and for the reason that, 

 in the long run, there must turn up as many 

 sixes as ones or twos or threes or fours or 

 fives. " The die has six sides, all of which 

 are turned up with equal frequency." Of 

 course, the greater the number of throws, 

 the nearer will the numbers of times which 

 each side of the die falls uppermost ap- 

 proximate each other approaching rela- 

 tively nearer all the way from six throws to 

 sixty million, and on to infinity. If a six 

 has failed to turn up for twenty throws, 

 and if it must turn up as frequently as the 

 other numbers, it must some time after the 

 twentieth throw make up the deficiency. 

 To average up, six must begin some time to 

 turn oftener ; that is, with each failure to 

 fall uppermost, its chance or probability of 

 doing so is increased. 



On the other hand, suppose you have 

 thrown the die twenty times, or twenty 

 thousand times, and have failed to turn a 

 six, even then the twenty-thousand-and-first 

 throw, considered by itself, manifestly af- 

 fords one-sixth (no more, no less) of a 

 chance of turning up a six. 



How is this logic to be reconciled ? 

 Respectfully, 



Charles West. 

 San Francisco, California, April 2, 1878. 



We insert this letter because it gives 

 expression to a fallacious notion which is 

 very current. At the gambling-places they 

 distribute cards upon which the players 

 can, by prickings, mark the number of times 

 which black and red have turned up, so as 

 to bet upon the color which is in deficiency. 

 The confusion is between the following two 

 statements, of which the first is true, the 

 second false : 



1. " If a die be thrown a sufficient num- 

 ber of times, the proportion of times with 

 which it will turn up six, will approximate 

 (within any desired limit) to one-sixth." 

 This is true. 



2. " If a die be thrown a sufficient num- 

 ber of times, the number of times with 

 which it will turn up six will approximate 

 (within any desired limit) to one-sixth of 

 the total number of throws." This is plain- 

 ly false. Suppose a die be thrown six 

 times in all, then the number of times in 

 which six comes up cannot differ by more 

 than five from being exactly one-sixth of 

 the total number of throws. But does any- 

 body imagine that, if it were thrown six 

 hundred times, the number of sixes would 

 often lie between 95 and 105, or within 

 five of one-sixth of the total number ? 



Recognizing this distinction, our cor- 

 respondent's argument falls at once to the 

 ground. Suppose that the first twenty 

 throws of the die were to be six, and there- 

 after just one-sixth of the throws were to 

 be six, then the frequency of sixes would be 

 as follows : 



After 20 throws, the frequency would 

 , 20 

 be 20= ! - 



After 80 throws, the frequency would 



be 



be 



be 



20 + 10 

 80 



= 0.375. 



After 620 throws, the frequency would 



20 + 100 



= 0.19354839. 



After 6,020 throws, the frequency would 



20 + 1,000 



= 0.1694352. 



6,020 



Thus the frequency would continually 

 approximate toward one-sixth or 0.166666 

 . . ., although sixes were thrown exactly 

 one-sixth of the time, after the run of twenty 

 sixes. Our correspondent is, therefore, in 

 error when he says : " If a six has failed to 

 turn up for twenty throws, and it must turn 

 up as frequently as the other numbers, it 

 must some time after the twentieth throw 

 make up the deficiency. To average up, 



