In an essay competition, the odds in favour of competitors P, Q, R and S are 1:2, 1:3, 1:4 and 1:5 respectively. Find the probability that one of them wins the competition.

Given : Probability that P wins  =  1/2  
Probability that Q wins  =  1/3
 Probability that R wins   =  1/4 
Probability that S  wins  = 1/5 thus
Probability that P loss =1- 1/2= 1/2  ,
Probability that Q loss  = 1 - 1/3   =  2/3 ,
Probability that R loss   = 1 -  1/4  =  3/4 and
Probability that  S loss  = 1 - 1/5   =  4/5  thus 
 Probability that one of them wins the comptitions,                         
  =  $(P * \mathrm{Q\text{'}} * \mathrm{R\text{'}} * S\text{'}) + (Q* \mathrm{P\text{'}} * \mathrm{R\text{'}} * \mathrm{S\text{'}} ) + (R * \mathrm{Q\text{'}} * \mathrm{P\text{'}} * \mathrm{S\text{'}} ) + (S * \mathrm{P\text{'}} * Q\text{'} * \mathrm{R\text{'}})$                 
                ( P wins and  Q, R, S LOSS  +  Q wins and P, R,S wins  +  R wins and P , Q, S loss  +  S wins and P,                                                                                                                                                                                   Q, R loss)

= (1/2 * 2/3 * 3/4 * 4/5)  + (1/3 * 1/2 * 3/4 * 4/5)  + (1/4 * 1/2 * 2/3 * 4/5)  + (1/5 * 1/2 * 2/3 * 3/4)                                 =1/5 + 1/10 + 1/15 + 1/20    
 =(12 + 6 + 4 + 3)/60             
 =25/60                     
 =5/12