Good Luck Chuck
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IMDB rating: 5.40 Plot: In order to keep the woman of his dreams from falling for another guy, Charlie Logan has to break the curse that has made him wildly popular with single women: Sleep with Charlie once, and the next man you meet will be your true love. |
Actors: Fogler Dan,Teigen Michael,Ayres Ben,Bacic Steve,Cook Dane,Gentile Troy,James Liam,Klop Cody,Price Connor,Ross Lonny,Volonino Daniel,Welch Ed,Comedy,Romance,
Can you help me solve this AP Statistics question?
The payoffs in the casino game Chuck-a-Luck are determined by the number of threes that appear when a set of four-sided dice are tossed.
A four-sided die has the numbers 1, 2, 3 and 4 on its faces.
a) If 20 dice are tossed, what is the probability that at least half will be threes?
b) If 10 dice are tossed, what is the probability that at most 4 threes will appear?
c) If a person rolls one four-sided die, he can win money if he does not roll a three any of the first five times he rolls. What’s the probability a person will win in this fashion?
a)
Let X be the number of threes appearing. X has the binomial distribution with n = 20 trials and success probability p = 0.25
In general, if X has the binomial distribution with n trials and a success probability of p then
P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)
for values of x = 0, 1, 2, …, n
P[X = x] = 0 for any other value of x.
The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.
Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.
X ~ Binomial( n = 20 , p = 0.25 )
the mean of the binomial distribution is n * p = 5
the variance of the binomial distribution is n * p * (1 - p) = 3.75
the standard deviation is the square root of the variance = v ( n * p * (1 - p)) = 1.936492
The Probability Mass Function, PMF,
f(X) = P(X = x) is:
P( X = 0 ) = 0.003171212
P( X = 1 ) = 0.02114141
P( X = 2 ) = 0.06694781
P( X = 3 ) = 0.1338956
P( X = 4 ) = 0.1896855
P( X = 5 ) = 0.2023312
P( X = 6 ) = 0.1686093
P( X = 7 ) = 0.1124062
P( X = 8 ) = 0.06088669
P( X = 9 ) = 0.02706075
P( X = 10 ) = 0.009922275
P( X = 11 ) = 0.00300675
P( X = 12 ) = 0.0007516875
P( X = 13 ) = 0.0001541923
P( X = 14 ) = 2.569872e-05
P( X = 15 ) = 3.426496e-06
P( X = 16 ) = 3.569266e-07
P( X = 17 ) = 2.799425e-08
P( X = 18 ) = 1.555236e-09
P( X = 19 ) = 5.456968e-11
P( X = 20 ) = 9.094947e-13
20
? P(X = t) =
t = 10
P( X ? 10 ) = 0.01386442
b )X ~ Binomial( n = 10 , p = 0.25 )
P( X = 0 ) = 0.05631351
P( X = 1 ) = 0.1877117
P( X = 2 ) = 0.2815676
P( X = 3 ) = 0.2502823
P( X = 4 ) = 0.145998
4
? P(X = t) =
t = 0
P( X ? 0 ) = 0.05631351
P( X ? 1 ) = 0.2440252
P( X ? 2 ) = 0.5255928
P( X ? 3 ) = 0.7758751
P( X ? 4 ) = 0.9218731
c)
first five rolls are non threes: (3/4) ^ 5 = 0.2373047
Merlyn | Jan 25, 2010