A bag contains 4 red, 5 blue and 6 green balls. The balls are indistinguishable except for their colour. A trial consists of drawing a ball at random from the bag, noting its colour and replacing it in the bag. A game is plated by performing 10 trials in all.

At the start of the tournament, each player plays the above game once. Players who earned more than $k proceed to the next round. Find the least value of k such that, in a random sample of 10 players, the probability that all 10 players proceed to the next round is less than 0.1.

Let X be the number of blue balls drew.

X~B(10,$\displaystyle \frac{1}{3}$)

$\displaystyle [P(X>n)]^{10} < 0.1$ where $\displaystyle n=\frac{k}{0.50}$

$\displaystyle 1-P(X $≤ $\displaystyle n) <0.794$

$\displaystyle P(X $≤ $\displaystyle n) > 0.206$