426 Prof. Forbes on the evidence for a Physical Connexion 



which may be written 



Y 1.2. ».p J_ 



1.2. ...(p — n) p n ' 

 We may apply this theorem to Mitchell's case of the star 

 /3 Capricorni, which consists (he states) of two stars within 3^- 

 minutes of one another, and there are in the sky 230 stars of 

 similar brightness. Now the area of a space 3^' radius is 

 4254603 less than the area of the entire sphere. Hence 

 p = 4254603 and n — 230. The val ue of the above expression, 

 carefully calculated by the aid of Stirling's theorem*, gives the 



chance for the proximity in question = '00617, or about — — . 



Mitchell finds it — (Phil. Trans, lvii. p. 246). The proce- 

 81 



dure of Mitchell where several stars are concerned — for in- 

 stance the Pleiades — seems to me to be still more deficient in 

 evidence ; and, in fact, I find from my mathematical friend, 

 that when the question is as to the chance of several such dice 

 showing at once the same face, the problem rises to an exces- 

 sive degree of complication, I conclude, therefore, that the 

 probability of 500,000 to 1 against the fortuitous concourse of 

 the Pleiades is, even granting all the premisses, erroneously 

 calculated. 



Note B. 



In thejirst supposed case, the probability of the compound 

 event of 100 dice turning up all the running numbers 1 ... 100 

 may be conceived to be the product of the independent proba- 

 bilities of 98 out of the 100 dice showing the numbers 1 ... 98, 

 or any other different numbers which are predetermined (we 



will call the probability of this ■=), and of the two remaining 



dice (having 100 faces each) showing the two faces 99 and 

 100, or the two remaining numbers. As this latter event may 

 happen two ways, the probability of it is 



< Too x Tob = 5000 : 



and TT7r;--p is the probability of the given compound event. 

 In the second supposed case, 98 numbers are,to be turned up 

 as before, with a probability -p. The two remaining ones are 



1/JL\ 



■ \P— n ) 



When n is much greater than 1, and p than n, this is almost equal to 



The probability is 1 _ 1 / p ^p-n+k 



n 2 



« is almost eoual to — . 



2p 



