What is the Least Value of K for Advancement in a Binomial Distribution Game?

[Punch]

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]

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]

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]

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):

            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

 

[CellCoree]

Using a binomial distribution table or a calculator, we can find that the probability of getting 5 or less blue balls out of 10 trials is approximately 0.206. Therefore, the least value of k would be 5 dollars, since a player would need to have at least 5 blue balls to proceed to the next round. This ensures that the probability of all 10 players proceeding to the next round is less than 0.1, as requested.