A particular game at an arcade in Philadelphia operates according to the followi

Question

A particular game at an arcade in Philadelphia operates according to the following rule: there are 40 boxes on the table, of which 16 are black, 16 are red, 4 are green, and 4 are yellow. A ball is thrown and the player can gain/lose money depending on where the ball lands. If the ball lands in a red box, the player will get 1 in cashable store credit; otherwise the player simply loses his/her 1 paid to play the game. Suppose a hopeful player throws the ball with equal probability of ending up in one of the boxes (assume here that the probability is zero that the ball ends up being in none of the boxes). Let X be the amount of money the player gains/loses, which can be a positive/negative amount, respectively. Specify the pmf of X, noting that it is a discrete RV since there are only limited possible outcomes. Find the mean and standard deviation of X. The Central Limit Theorem tells us the approximate distribution of X50, the mean amount of money the player gains/loses if s/he plays 50 times. What is this distribution? Use the empirical rule for normal distribution to give the range in which X50 will fall 95% of the time. Multiply by 50 to get the middle 95% of the distribution of the amount of money (i.e., not the mean) gained/lost when the player plays 50 times. What is the probability that the player will lose money if s/he plays 50 times? (This is the probability that 50 is less than 0). If 100.000 such games are played at the arcade in a year, what is the distribution of 100,000?

Explanation / Answer

p for landing in red box=16/40

a)

X= 1 if red box 16/40

1 if other box 24/40

b)mean=x*p=16/40-24/40= -0.2

sd=sqrt(x^2*p)=16/40+24/24=1

can also answer the rest

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