CPE101 Project 3 Solved

Project3

#Our project is based on the football with a slightly below average team such as the Da #so we will investigate the streaks of winning and losing in a season.

In [5]: from symbulate import *

%matplotlib inline

In [6]: #returns the largerst success streak with counter variables def count_streak_of_success(omega):

streak = 0 curmax = 0 for i, w in enumerate(omega): if w == 0:

if curmax < streak:

curmax = streak

streak = 0

if w == 1: streak += 1 return curmax

#returns the largest losing streak with counter variables def count_streak_of_failure(omega):

streak = 0 curmax = 0 for i, w in enumerate(omega): if w == 1:

if curmax < streak:

curmax = streak

streak = 0

if w == 0: streak += 1 return curmax #returns the larger of the 2 streaks with counter variables def max_streak(omega):

streak_0 = 0 streak_1 = 0 curmax = 0 for i, w in enumerate(omega): if w == 1: if curmax < streak_1:

curmax = streak_1

streak_1 = 0 streak_0 += 1

if w == 0: if curmax < streak_0:

curmax = streak_0

streak_1 += 1 streak_0 = 0

return curmax

#returns amount of times success(win) streak failure(losing) streak def prob_streak(omega):

streak_0 = 0 streak_1 = 0 curmax = 0 for i, w in enumerate(omega): if w == 1: if curmax < streak_1:

curmax = streak_1

streak_1 = 0 streak_0 += 1

if w == 0: if curmax < streak_0: curmax = streak_0

streak_1 += 1

streak_0 = 0

if curmzx_0 curmax_1:

return True

else: return False

In [7]: #1a) Distribution of X with P(success)=.45

P = BoxModel([1, 0], probs=[.45, .55], size=16) x = count_streak_of_success

X = RV(P, x)

X.sim(10000).plot()

X.sim(10000).tabulate()

Out[7]: {2: 3501, 4: 1423,

3: 2902,

1: 1077,

9: 11,

5: 653,

6: 259,

10: 7,

7: 111,

12: 4,

8: 47,

0: 3,

11: 2}

 

In [8]: #1a) Distribution of Y with P(Failure or Losing) = .55

P = BoxModel([1, 0], probs=[.45, .55], size=16) y = count_streak_of_failure

Y = RV(P, y)

Y.sim(10000).plot()

Y.sim(10000).tabulate()

Out[8]: {3: 2975,

5: 1184,

4: 2015,

2: 2305,

7: 280,

6: 577,

1: 378,

8: 152,

9: 68,

10: 39,

12: 9,

13: 1,

0: 1,

11: 13,

14: 3}

 

In [9]: #1a) Distribution of Z

P = BoxModel([1, 0], probs=[.45, .55], size=16) z = max_streak Z = RV(P, z)

Z.sim(10000).plot()

 

In [10]: #1b)
P = BoxModel([1, 0], probs=[.45, .55], size=16) x = count_streak_of_success X = RV(P, x)

y = count_streak_of_failure Y = RV(P, y) xy = (X & Y).sim(10000)

xy.plot(['tile','joint']) xy.cov()

count = 0 for i in xy:

if i[0] < i[1]:

count += 1
                    1 - count/10000                          #Equal to the probability that X Y

Out[10]: 0.4766

 

In [11]: #1c)
P = BoxModel([1, 0], probs=[.45, .55], size=16) x = count_streak_of_success X = RV(P, x) z = max_streak Z = RV(P, z) xz = (X & Z).sim(10000)

xz.plot(['tile','joint']) xz.cov()

#The probability that X Z is 0 because X can only equal Z at most in cases where X

Out[11]: 0.60700265026502598

 

In [12]: #Number 2

P1 = BoxModel([1, 0], probs=[.30, .70], size=16) x1 = count_streak_of_success

X1 = RV(P, x1)

X1.sim(10000).plot()

P2 = BoxModel([1, 0], probs=[.50, .50], size=16) x2 = count_streak_of_success

X2 = RV(P, x2)

X2.sim(10000).plot(jitter = True)

P3 = BoxModel([1, 0], probs=[.70, .30], size=16) x3 = count_streak_of_success

X3 = RV(P, x3)

X3.sim(10000).plot(jitter = True)

#There are multiple graphs overlayed but it is hard to see unless you zoom in.

 

In [13]: #Number 2 - Covariance with P(Success) = .30

P1 = BoxModel([1, 0], probs=[.30, .70], size=16) x1 = count_streak_of_success

X1 = RV(P, x1)

X1.sim(10000).plot() y = count_streak_of_failure Y = RV(P, y) x1y = (X1 & Y).sim(10000) x1y.plot(['tile','joint'])

for i in x1y:

if i[0] < i[1]:

count += 1
X1.sim(10000).mean(), X1.sim(10000).var(), 1 - count/10000, x1y.cov()

Out[13]: (2.895, 1.8394519900000004, -0.057399999999999896, -0.63711507150715063)

 

In [14]: #Number 2 - Covariance with P(Success) = .50

P1 = BoxModel([1, 0], probs=[.50, .50], size=16) x1 = count_streak_of_success

X1 = RV(P, x1)

X1.sim(10000).plot() y = count_streak_of_failure Y = RV(P, y) x1y = (X1 & Y).sim(10000) x1y.plot(['tile','joint'])

for i in x1y:

if i[0] < i[1]:

count += 1

X1.sim(10000).mean(), X1.sim(10000).var(), 1 - count/10000, x1y.cov()

Out[14]: (2.897, 1.8779267899999998, -0.5828, -0.68454853485348477)

 

In [15]: #Number 2 - Covariance with P(Success) = .70

P1 = BoxModel([1, 0], probs=[.70, .30], size=16) x1 = count_streak_of_success

X1 = RV(P, x1)

X1.sim(10000).plot() y = count_streak_of_failure Y = RV(P, y) x1y = (X1 & Y).sim(10000) x1y.plot(['tile','joint'])

for i in x1y:

if i[0] < i[1]:

count += 1

X1.sim(10000).mean(), X1.sim(10000).var(), 1 - count/10000, x1y.cov()

Out[15]: (2.8859, 1.8253958399999999, -1.1027, -0.6260349734973496)

 

In [16]: #Number 3

P = BoxModel([1, 0], probs=[.45, .55], size=16) x = count_streak_of_success

X = RV(P, x)

X.sim(10000).plot()

Q = BoxModel([1, 0], probs=[.45, .55], size=160) y = count_streak_of_success

Y = RV(P, y)

Y.sim(10000).plot(jitter = True)

R = BoxModel([1, 0], probs=[.45, .55], size=1600) z = count_streak_of_success

Z = RV(P, z)

Z.sim(10000).plot(jitter = True)

 

In [17]: #Number 3 - Covariance with size = 16

P = BoxModel([1, 0], probs=[.45, .55], size=16) x = count_streak_of_success

X = RV(P, x)

X.sim(10000).plot() y = count_streak_of_failure Y = RV(P, y) x1y = (X & Y).sim(10000) x1y.plot(['tile','joint'])

for i in x1y:

if i[0] < i[1]:

count += 1

X.sim(10000).mean(), X.sim(10000).var(), 1 - count/10000, x1y.cov()

Out[17]: (2.8866, 1.81457024, -1.6353, -0.68238275827582717)

 

In [18]: #Number 3 - Covariance with size = 160

Q = BoxModel([1, 0], probs=[.45, .55], size=160) x = count_streak_of_success

X = RV(P, y)

X.sim(10000).plot() y = count_streak_of_failure Y = RV(P, y) x1y = (X & Y).sim(10000) x1y.plot(['tile','joint'])

for i in x1y:

if i[0] < i[1]:

count += 1

X.sim(10000).mean(), X.sim(10000).var(), 1 - count/10000, x1y.cov()

Out[18]: (3.6061, 2.87357751, -1.6353, 2.7956305630563061)

 

In [19]: #Number 3 - Covariance with size = 1600

R = BoxModel([1, 0], probs=[.45, .55], size=1600) x = count_streak_of_success

X = RV(P, y)

X.sim(10000).plot() y = count_streak_of_failure Y = RV(P, y) x1y = (X & Y).sim(10000) x1y.plot(['tile','joint'])

for i in x1y:

if i[0] < i[1]:

count += 1

X.sim(10000).mean(), X.sim(10000).var(), 1 - count/10000, x1y.cov()

Out[19]: (3.6266, 2.76439136, -1.6353, 2.7551282228222798)

 

In [20]: #4
#We will now investigate how many games it takes a team to win 5 games in a row with a

#of 80%. If the team never wins 5 games in a row, it gets coded as 0

r = 5

def count_until_rth_streak(omega):

count = 0 streak = 0 curmax = 0 for i, w in enumerate(omega): if w == 0:

if curmax < streak: curmax = streak count += 1 streak = 0 count += 1

if w == 1:
streak += 1

count += 1
if streak == 5:

return count return 0

P = BoxModel([1, 0], probs=[0.80, 0.20], size=16)

X = RV(P, count_until_rth_streak) x = X.sim(10000) x.plot()

#With a probability of .80 of winning, it takes on average 6-7 games to win 5 in a row

 

In [21]: #4
#We will now investigate how many games it takes a team to win 5 games in a row with a

#of 50%. If the team never wins 5 games in a row, it gets coded as 0

r = 5 def count_until_rth_streak(omega):

count = 0 streak = 0 curmax = 0 for i, w in enumerate(omega): if w == 0:

if curmax < streak: curmax = streak count += 1 streak = 0 count += 1

if w == 1:

streak += 1

count += 1

if streak == 5: return count return 0

P = BoxModel([1, 0], probs=[0.5, 0.5], size=16) X = RV(P, count_until_rth_streak) x = X.sim(10000) x.plot()

#With a probability of .5 of winning, it takes on average 10-12 games to win 5 in a ro

#that more often than not, the team never reaches a win streak of 5 games

 

In [22]: #4
#We will now investigate how many games it takes a team to win 5 games in a row with a

#of 65%. If the team never wins 5 games in a row, it gets coded as 0

r = 5 def count_until_rth_streak(omega):

count = 0 streak = 0 curmax = 0 for i, w in enumerate(omega): if w == 0:

if curmax < streak:

curmax = streak count += 1 streak = 0 count += 1

if w == 1: streak += 1 count += 1

if streak == 5:

return count return 0

P = BoxModel([1, 0], probs=[0.65, 0.35], size=16) X = RV(P, count_until_rth_streak) x = X.sim(10000) x.plot()

#With a probability of .65 of winning, it takes on average 8-9 games to win 5 in a row

#that more often than not, the team never reaches a win streak of 5 games but a little

#in the distribution of win streaks than in the previous graph

 

In [23]: #4
#We will now investigate how many games it takes a team to win 5 games in a row out of

#rather than 16 with a win probability of 65%. If the team never wins 5 games in a row

r = 5 def count_until_rth_streak(omega):

count = 0 streak = 0 curmax = 0 for i, w in enumerate(omega): if w == 0:

if curmax < streak: curmax = streak count += 1 streak = 0 count += 1

if w == 1:

streak += 1

count += 1

if streak == 5:

return count

return 0

P = BoxModel([1, 0], probs=[0.65, 0.35], size=48) X = RV(P, count_until_rth_streak) x = X.sim(10000) x.plot()

#With a probability of .65 of winning and a total of 48 games, it takes on average 25 #We see that in the simulation a team will win 5 games in a row more times than not.