Probability Question 223
Question: A is one of 6 horses entered for a race, and is to be ridden by one of two jockeys B and C. it is 2 to 1 that B rides A, in which case all the horses are equally likely to win. If C rides A, his chance of winning is trebled. What are the odds against winning of A?
Options:
A) $ 5:13 $
B) $ 5:18 $
C) $ 13:5 $
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
Let E= the event that horse A wins $ E_1= $ the event that jockey B rides hors A $ E_2= $ the event that jockey C rides hors A According to question odds in favour of $ E_1=2:1 $
$ \therefore P(E_1)=\frac{2}{3} $
and $ P( \frac{E}{E_1} )=\frac{1}{6} $ (Since, when B rides A, all six
Horses are equally likely to win)
$ P(E_2)=1-P(E_1)=1-\frac{2}{3}=\frac{1}{3} $
and $ P( \frac{E}{E_2} )=3P( \frac{E}{E_1} )=\frac{1}{2} $
Let $ A_1=E_1\cap E $ and $ A_2=E_2\cap E $
Now, required probability
$ P(E)=P(A_1)+P(A_2) $
$ =p(E_1\cap E)+P(E_1\cap E) $
$ =p(E_1)P( \frac{E}{E_1} )+P(E_2)P( \frac{E}{E_2} ) $
$ =\frac{2}{3}.\frac{1}{6}+\frac{1}{3}.\frac{1}{2}=\frac{5}{18}. $
So, that odds against winning of A are 13 : 5.
BETA
