Answer: ![\dfrac{26}{27}](https://tex.z-dn.net/?f=%5Cdfrac%7B26%7D%7B27%7D)
Step-by-step explanation:
Given : To avoid detection at customs, a traveler places 6 narcotic tablets in a bottle containing 9 vitamin tablets that are similar in appearance.
Proportion of success : ![p=\dfrac{6}{9}=\dfrac{2}{3}](https://tex.z-dn.net/?f=p%3D%5Cdfrac%7B6%7D%7B9%7D%3D%5Cdfrac%7B2%7D%7B3%7D)
Sample size taken by customs official : n= 3
Let x be a binomial variable that represents the tablets in the bottle.
Using Binomial probability formula :-
![P(X=x)=^nC_xp^x(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%3Dx%29%3D%5EnC_xp%5Ex%281-p%29%5E%7Bn-x%7D)
The probability that the traveler will be arrested for illegal possession of narcotics =![P(x\geq1)=1-P(x=0)](https://tex.z-dn.net/?f=P%28x%5Cgeq1%29%3D1-P%28x%3D0%29)
![1-^3C_0(\dfrac{2}{3})^0(1-\dfrac{2}{3})^3\\\\=1-(1)(1)(\dfrac{1}{3})^3\ [\becuase ^nC_0=1]\\\\=1-\dfrac{1}{27}=\dfrac{26}{27}](https://tex.z-dn.net/?f=1-%5E3C_0%28%5Cdfrac%7B2%7D%7B3%7D%29%5E0%281-%5Cdfrac%7B2%7D%7B3%7D%29%5E3%5C%5C%5C%5C%3D1-%281%29%281%29%28%5Cdfrac%7B1%7D%7B3%7D%29%5E3%5C%20%5B%5Cbecuase%20%5EnC_0%3D1%5D%5C%5C%5C%5C%3D1-%5Cdfrac%7B1%7D%7B27%7D%3D%5Cdfrac%7B26%7D%7B27%7D)
Hence, the probability that the traveler will be arrested for illegal possession of narcotics = ![\dfrac{26}{27}](https://tex.z-dn.net/?f=%5Cdfrac%7B26%7D%7B27%7D)