The chance of one event happening is the square of the chance of a second event, but the odds against the first are cube of the odds against the second. The chance of the first event is

Question

The chance of one event happening is the square of the chance of a  second event, but the odds against the first are cube of the odds  against the second. The chance of the first event is

Infinity Learn · Accepted Answer

Let the two events be $$E1$$     $$andE2$$     . Let their chances be p and q respectively.   Then $$p=q2$$     …………….(i)  The chances of not happening  of the events are $$1$$−p     and $$1$$−q     respectively.Odds against the first event $$=$$ $$1$$−pp     Odds against the second event $$=$$ $$1$$−qq     Given $$1$$−pp      $$=(1$$−$$qq)3$$⇒$$1$$−$$q2q2=(1$$−$$q)3q3$$[from   (i)]     ⇒(1$$−$$qq2)[(1+q)−(1$$−$$q)2q$$]$$=0$$      ∵q≠$$1$$  and  q≠$$0$$     ∴$$q(1+q)=1$$−$$2q+q2$$⇒$$q=13$$     ∴     from $$(i) $$p=q2=19$$     ∴$$p(E1)=p=19$$

The chance of one event happening is the square of the chance of a second event, but the odds against the first are cube of the odds against the second. The chance of the first event is

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