# Casino Questions:

#1.  (Game="Silly Dice": The player pays 1 dollar.  A six ice is rolled. If a six comes the player  get 4 dollars!) Imagine, this game is played by 50 people a day.  What is the probability that more than 15 people  will win on a given day?


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#Solution:  Since (1/6)*50<10, we can not use the normal approximation.

p=1/6
q=1-p
N=50
k=0:15
Prob=1-sum(dbinom(k,N,p))
Prob

# In this case one could use the factorials without any thought and hecne the "standard" formula for the binomial coefficients. 

W=(p^(k))*q^(N-k)
X=factorial(k)
Y=factorial(N-k)
Prob=1-sum((factorial(N)/(X*Y))*W)
Prob
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# 2. (Game=Roulette:  The player pays 1 dollar.  A wheel with 38 slots is spun.   If a 00 arises (which is one of the 38 slots)  the player gets 35 dollars!)
Imagine, this game is played by 1,000 people a week.    What is the probability that   20 or fewer people will win on a given week?

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#Solution: 

p=1/38
q=1-p
N=1000
k=0:20
Prob=sum(dbinom(0:20,N,p))
Prob

#Solution 2:  Since (1/38)*1000>=10, we can also use the normal approximation.

z=(20.5-N*p)/(sqrt(p*q*N))
ProbApprox=pnorm(z,0,1)
ProbApprox


#Notice the 0.5 in 20.5.  

#We cannot sue the "standard" formula in the naive sense, try

W=(p^(k))*q^(N-k)
X=factorial(k)
Y=factorial(N-k)
Prob=1-sum((factorial(N)/(X*Y))*W)
Prob

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#3.  (Game=Crazy Jelly Bean "Over 100 million dollars in prizes!: and over 30000 chances to win! The player pays 1 dollar.  While blindfolded, a player uses a grabbing device to pick a jelly bean out of a swimming pool that contains roughly 280,000,000 jelly beans.  30,800 of these jelly beans are   filled with gold! and can be exchanged for a check for 5,000 dollars! ) Imagine, this game is played by 35000 people a decade.  What is the probability that more than 7 people  win in a given decade?

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# Solution: 


p=30800/(280*10^6)
q=1-p
N=35000
k=0:7
Prob=1-sum(dbinom(k,N,p))
Prob

#Solution 2: Since, N>>k, and N>>N*p, we can use the Poisson Approximation:

l=N*p
X=l^k
Y=factorial(k)
ProbApprox=1-sum((X/Y)*(exp(-l)))
ProbApprox

# Once again, ewe cannot sue the "standard" formula in the naive sense.