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

AI Thread Summary
In a binomial distribution game involving a bag of colored balls, players draw balls over 10 trials, earning $0.50 for each blue ball drawn. The goal is to determine the least value of k such that the probability of all 10 players advancing, having earned more than k dollars, is less than 0.1. The random variable X, representing the number of blue balls drawn, follows a binomial distribution X~B(10, 1/3). Calculations show that the smallest number of blue balls needed for the probability P(X ≤ n) to exceed 0.206 is n=2, leading to the conclusion that k must be set accordingly. The analysis concludes that k must be adjusted to ensure that the advancement probability criterion is met.
Punch
Messages
44
Reaction score
0
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 $≤ $n) > 0.206$
 
Mathematics news on Phys.org
Punch said:
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
 
CaptainBlack said:
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
 
Punch said:
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
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top