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Question

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(a) 25 to 8

(b) 27 to 8

(c) 4 to 1

(d) 27 to 35

Answer

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* Odds a to b represent that the probability of champion ship is either $\dfrac{a}{{a + b}}$ or $\dfrac{b}{{a + b}}$ depending upon the person whose probability is to be found.

* Probability of an event =Number of favourable event/Total event

* Probability of either event1 or event2 = probability of event1 + probability of event2. Where these events are disjoint which means the probability of happening of event1 and event2 is zero.

Representing given information in probability form:

We use the formula $\dfrac{a}{{a + b}}$

Odds of A, 4 to 3 represents that a = 4 and b = 3 , so a+b = 4+3 = 7

Probability (A becomes the champion),

\[P(A) = \dfrac{4}{7}\]

Odds of B, 1 to 4 represents a = 1 and b = 4 , so a+b = 1+4 = 5

Probability (B becomes the champion),

\[P(B) = \dfrac{1}{5}\]

Probability that either A or B will become the champion

= P(A becomes champion) + P(B becomes champion.)

$P(A) + P(B) = \dfrac{4}{7} + \dfrac{1}{5}$

Take LCM on right hand side of the equation

$

P(A) + P(B) = \dfrac{{4 \times 5 + 1 \times 7}}{{7 \times 5}} \\

P(A) + P(B) = \dfrac{{27}}{{35}} \\

$

These two events are disjoint. As happening in both events which are both A and B wins have probability zero.

Converting probability to odds.

Comparing the value \[\dfrac{{27}}{{35}}\] to $\dfrac{a}{{a + b}}$, we get \[a = {\text{ }}27,a + b = 35,\]

\[ \Rightarrow b = 35 - a \Rightarrow b = 35 - 27 = 8\]

So, a = 27 and b = 8

Odds that A or B will become the champion is 27 to 8