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**Stats** Please let me know where if i am right or no?

Answer:

Binomial P(x;n,$\displaystyle \theta$) = $\displaystyle \binom{n}{x} \theta ^x (1- \theta)^{n-x})$

Probability of atleast 1 success is

$\displaystyle 1-P(X=0) $ = $\displaystyle 1 -\binom{n}{0} \theta ^0 (1- \theta)^{n-0}$ = $\displaystyle 1 - (1 - \theta)^n$

Probability of k success occuring in first n trials is

$\displaystyle P(X=k) = \binom{n}{k} \theta^k (1- \theta)^{n-k}$

Probability of all trials being success is

$\displaystyle P(X=n) = \binom{n}{n} \theta^n (1- \theta)^{n-n} =\theta^n$