# Binomial distribution

#### Punch

##### New member
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 dollars 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,$\frac{1}{3}$)

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

$1-P(X$≤ $n) <0.794$

$P(X$≤ $n) > 0.206$

#### CaptainBlack

##### Well-known member
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 dollars 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,$\frac{1}{3}$)

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

$1-P(X$≤ $n) <0.794$

$P(X$≤ $n) > 0.206$
Incomplete question. Please include all the relevant information to the question in the thread with the question.

CB

#### Punch

##### New member
Incomplete question. Please include all the relevant information to the question in the thread with the question.

CB
Sorry! The missing part is: For each blue ball obtained, the player earns $0.50 #### CaptainBlack ##### Well-known member Sorry! The missing part is: For each blue ball obtained, the player earns$0.50
OK, so make a table of b(i,10,1/3):

Code:
            i     b(i,10,1/3)
----------------
0     0.0173415
1     0.0867076
2      0.195092
3      0.260123
4      0.227608
5      0.136565
6     0.0569019
7     0.0162577
8    0.00304832
9   0.000338702
10  1.69351e-005
Now you need another column with the cumulative sum ...

(n=2 is the smallest number of wins such that P(X<=n)>0.206)

CB