help with geometric distribution exercise

Aug 2011
5
0
Hello dear forum members. I was requested to answer this exercise that requires the use of geometric distribution. "A bag contains 3 white and 5 black marbles. A marble is extracted with replacement until a black marble is selected. Find the probability that at least k-extractions are required".

I need some help with the procedure to find the final answer.

I have: P(X>=K)=1-P(X<K)=1-[sum from i=1 until k-1] (3/8)^(i-1)*5/8

The final answer provided is (3/8)^(k-1), but I'm lost in the intermediate steps.

Any help is greatly appreciated.
 

Plato

MHF Helper
Aug 2006
22,462
8,634
Hello dear forum members. I was requested to answer this exercise that requires the use of geometric distribution. "A bag contains 3 white and 5 black marbles. A marble is extracted with replacement until a black marble is selected. Find the probability that at least k-extractions are required".
I need some help with the procedure to find the final answer. The final answer provided is (3/8)^(k-1)
If $X$ is the number of the draw upon which the first black ball appears, then there have appeared $X-1$ white balls.
Thus $\mathcal{P}(X=K)=\left(\dfrac{3}{8}\right)^{K-1}\left(\dfrac{5}{8}\right)$
So the complement,$1-$, is the probability of at least $\bf K$.