X B(33,p), 3P(X=0)=P(X=1).

Question

$X\sim B(33,p)$, $3P(X=0)=P(X=1)$. Find $\dfrac{P(X=15)}{P(X=18)}-\dfrac{P(X=16)}{P(X=17)}$.

Accepted Answer

From $3P(0)=P(1)$: $3q^{33}={}^{33}C_1 pq^{32}\Rightarrow 3q=33p\Rightarrow p=\dfrac{q}{11}$; with $p+q=1$, $q=\dfrac{11}{12},p=\dfrac{1}{12}$.

Using ratios of consecutive binomial terms, ${\dfrac{P(k)}{P(k+1)}}=\dfrac{(k+1)q}{(n-k)p}$, the expression evaluates to $1320$.

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