Question

three horses A, B and C are in race. A is twice as likely to win as B and B is twice as likely to win as C. What is their probabilities?

Answer 1

$\underline{\underline{\Huge{\textbf{Solution:}}}}$

Given,
3 Horses A, B, C are in a Race.

$\underline{\textit{Statement-1:}}$
A is twice as likely to win as B

$\underline{\textit{Statement-2:}}$
B is twice as likely to win as C



$\underline{\huge{\textsf{Step-1:}}}$
Assume a Variable to denote probability of Winning of Horse C

Let 'p' be the probability that Horse C wins the race. ————[1]



$\underline{\huge{\textsf{Step-2:}}}$
Rewrite Statement-2

Probability that Horse B wins the Race = 2(Probability that Horse C wins the Race)

Substitute [1]

$\implies$ Probability that Horse B wins the Race = 2p ————[2]



$\underline{\huge{\textsf{Step-3:}}}$
Rewrite Statement-1

Probability that Horse A wins the Race = 2(Probability that Horse B wins the Race)

Substitute [2]

$\implies$ Probability that Horse B wins the Race = 2(2p) = 4p ————[3]



$\underline{\huge{\textsf{Step-4:}}}$
Using the Property $\mathbf{P(S) = 1}$$\textsf{\: (Sum of Probabilities is \large{1})}$

We have,
S = A + B + C
P(S) = P(A) + P(B) + P(C)
$\therefore P(A) + P(B) + P(C) = 1$

Substitute [1] , [2] & [3]

$\implies 4p + 2p + 2p = 1$

$\implies 7p = 1$

$\implies p = \dfrac{1}{7}$



$\underline{\huge{\textsf{Step-5:}}}$
Find the Probabilities using [1], [2] & [3]

$P(C) = \dfrac{1}{7}$

$P(B) = 2(\dfrac{1}{7}) =\dfrac{2}{7}$

$P(B) = 2(\dfrac{2}{7}) =\dfrac{4}{7}$


$\therefore$
$\blacksquare \: \textsf{Probabilities of Winning of Horses A, B and C are \underline{\large{\textbf{ \dfrac{4}{7} \: , \dfrac{2}{7} \: and\: \dfrac{1}{7} }}}}$