Answer to Question #233556 in Statistics and Probability for Rashi

Question

Among the 16 cities that a professional society is considering for its next 3 annual conversation, 7 are in western part of the United States. To avoid arguments, the selection is left to chance. If none of the cities can be choosen more than once what are the probabilities that (a) none of the conversation will held western part of the United States? (b) all of the conversation will be held in western part of the United States?

Answer 1

Let $X=$ the number of of the conversation: $X\sim Bin(n, p).$

Given $n=3,p=\dfrac{7}{16},q=1-\dfrac{7}{16}=\dfrac{9}{16}$

(a)

$$P(X=0)=\dbinom{3}{0}(\dfrac{7}{16})^0(\dfrac{9}{16})^{3-0}=0.177978515625$$

$$\approx0.1780$$

(b)

$$P(X=3)=\dbinom{3}{3}(\dfrac{7}{16})^3(\dfrac{9}{16})^{3-3}=0.083740234375$$

$$\approx0.0837$$