import numpy as np
import matplotlib.pyplot as plt


## Why Probability and Statistics

• Why should you care about prob&stat?

• Certainty: Find the shortest route from a to b
• Uncertainty: Find the fastest route from a to b
• Search engine:

• Certainty: Find all web pages that contain the words "Trump", "Hillary", and "debate"
• Uncertainty: Find the 10 most relevant pages for the query "Trump, Hillary debate"
• Insurance Company:

• If the other side has the same color, you pay me \$1. ### Why is this a fair bet? • Suppose$U$us R. • Then, the card is either RR or RB. • Therefore, the other side can be either R or B. • Therefore, in this case, the odds are equal. • A similar argument holds for the case where$U$is B ### Let's use a monte-carlo simulation from random import random red_bck="\x1b[41m%s\x1b[0m" blue_bck="\x1b[44m%s\x1b[0m" red=red_bck%'R' black=blue_bck%'B' Cards=[(red,black),(red,red),(black,black)] counts={'same':0,'different':0} for j in range(50): # Select a random card i = int(random() * 3.) side = int(random() * 2.) C = Cards[i] # Select which side to be "up" if side == 1: C = (C[1], C[0]) same = 'same' if C[0] == C[1] else 'different' # Count the number of times the two sides are the same or different. counts[same] += 1 print(''.join(C) + ' %-9s' % same, end='') if(j + 1) % 5 == 0: print() print() print(counts)  BB same BB same RB differentBR differentRR same BB same BR differentRR same RR same RB different RR same BR differentRR same RR same RR same RR same RR same RR same RR same BR different BB same BR differentBB same RR same RR same BB same BB same BB same RR same BR different BB same RR same BB same BB same BB same RR same RR same RB differentRR same BB same RR same RR same RR same BB same RR same BB same RB differentRR same BB same RR same {'same': 40, 'different': 10}  ### The simulation does not agree with the argument • In simulation: the two sides have the same color about twice the number of times that they have different color. • You are twice as likely to lose as you are to win. • On Average, you lose 33 cents per iteration: $$\1 \times (2/3) - \1 \times (1/3)$$ ### Alternative argument If we pick a card at random 2/3 of the time, we pick a card where the two sides have the same color, and only 1/3 where the color is different. ### How can we be sure? • The original argument also sounds convincing, but is wrong. • To be sure that our argument is correct, we need to define some concepts, including outcome and event. ## Exercise In this excercise you will write code to estimate the probability that$n$flips of a fair coin will result in number of "heads" between$k_1$and$k_2$. You should write the body of two functions: 1. seq_sum(n): generates a random sequence of coin flips and counts the number of heads. 2. estimate_prob(n,k1,k2,m): Using calls to seq_sum, estimate the probability of the number of heads being between$k_1$and$k_2$. np.random.rand()  0.020644852872238384 np.random.rand(4)  array([0.95931166, 0.52807626, 0.08801368, 0.64187303]) ### Exercise 1 ​ Write a function, seq_sum(n), which generates$n$random coin flips from a fair coin and then returns the number of heads. A fair coin is defined to be a coin where$P($heads$)=\frac{1}{2}$​ The output type should be a numpy integer, hint: use np.random.rand() * **Code:** * x = seq_sum(100) print x print [seq_sum(2) for x in range(20)]  ​ ​ * **Output:** * 49 [0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 1] def seq_sum(n): """ input: n, generate a sequence of n random coin flips output: return the number of heads Hint: For simplicity, use 1,0 to represent head,tails """ seq_list = np.array([np.random.rand() for _ in range(n)]) return np.sum(seq_list >= 0.5)  x = seq_sum(100) print(x) assert np.unique([seq_sum(2) for x in range(0,200)]).tolist() == [0, 1, 2]  47  ### Exercise 2 Write a function, estimate_prob(n,k1,k2,m), that uses seq_sum(n) to estimate the following probability: $$P(\; k_1 <= \text{number of heads in n flips} < k_2 )$$ The function should estimate the probability by running$m$different trials of seq_sum(n), probably using a for loop. In order to receive full credit estimate_prob MUST call seq_sum (aka: seq_sum is located inside the estimate_prob function) * **Code:** * x = estimate_prob(100,45,55,1000) print(x) print type(x)  * **Output:** * 0.686 <type 'float'> def estimate_prob(n,k1,k2,m): """Estimate the probability that n flips of a fair coin result in k1 to k2 heads n: the number of coin flips (length of the sequence) k1,k2: the trial is successful if the number of heads is between k1 and k2-1 m: the number of trials (number of sequences of length n) output: the estimated probability """ success = 0 for _ in range(m): seqs = seq_sum(n) if k1 <= seqs < k2: success += 1 return success / m  x = estimate_prob(100,45,55,1000) print(x) assert 'float' in str(type(x))  0.688  ### Estimate vs. True Probability We can now check how to see how close these estimates are to the true probabilities. ### Helper Functions These helper functions are used to calculate the actual probabilities. They are used to test your code. It is not required that you understand how they work. def calc_prob(n,k1,k2): """Calculate the probability using a normal approximation""" n=float(n);k1=float(k1);k2=float(k2) z1=(k1-0.5*n)/(sqrt(n)/2) z2=(k2-0.5*n)/(sqrt(n)/2) return (erf(z2/sqrt(2))-erf(z1/sqrt(2)))/2 from math import erf,sqrt def evaluate(n,q1,q2,m,r=100): """Run calc_range many times and test whether the estimates are consistent with calc_prob""" k1=int(q1*n) k2=int(q2*n) p=calc_prob(n,k1,k2) std=sqrt(p*(1-p)/m) print('computed prob=%5.3f, std=%5.3f'%(p,std)) L=[estimate_prob(n,k1,k2,m) for i in range(r)] med=np.median(L) print('ran estimator %d times, with parameters n=%d,k1=%d,k2=%d,m=%d'%(r,n,k1,k2,m)) print('median of estimates=%5.3f, error of median estimator=%5.3f, std= %f5.3'%(med,med-p,std)) return L,med,p,std,abs((med-p)/std)  def test_report_assert(n,q1,q2,m,r=100): k1=int(q1*n) k2=int(q2*n) L,med,p,std,norm_err=evaluate(n,q1,q2,m,r=100) plt.hist(L); plt.plot([p,p],plt.ylim(),'r',label='true prob') plt.plot([med,med],plt.ylim(),'k',label='median of %d estimates'%r) mid_y=np.mean(plt.ylim()) plt.plot([p-std,p+std],[mid_y,mid_y],'g',label='+-std') plt.legend(); print('normalized error of median=',norm_err,'should be <1.0') plt.title('r=%d,n=%d,k1=%d,k2=%d,m=%d,\nnorm_err=%4.3f'%(r,n,k1,k2,m,norm_err)) assert norm_err<1.0  ### Testing your Functions • We now test your functions. The graphs below show how close your estimated probability is to the true probability for various values of$k_1$and$k_2\$. You can see that your answer is never exactly the correct probability.
m=100
i=1
plt.figure(figsize=[10,12])
for n in [100,1000]:
for q1,q2 in [(0.4,0.6),(0.55,1.00),(0.47,0.499)]:
fig=plt.subplot(3,2,i)
print('#### test no.',i)
i+=1
test_report_assert(n,q1,q2,m,r=100)
plt.tight_layout()

#### test no. 1
computed prob=0.954, std=0.021
ran estimator 100 times, with parameters n=100,k1=40,k2=60,m=100
median of estimates=0.950, error of median estimator=-0.004, std= 0.0208405.3
normalized error of median= 0.21591965634481614 should be <1.0
#### test no. 2
computed prob=0.159, std=0.037
ran estimator 100 times, with parameters n=100,k1=55,k2=100,m=100
median of estimates=0.185, error of median estimator=0.026, std= 0.0365355.3
normalized error of median= 0.7210739298262482 should be <1.0
#### test no. 3
computed prob=0.146, std=0.035
ran estimator 100 times, with parameters n=100,k1=47,k2=49,m=100
median of estimates=0.130, error of median estimator=-0.016, std= 0.0353595.3
normalized error of median= 0.46627417809069077 should be <1.0
#### test no. 4
computed prob=1.000, std=0.000
ran estimator 100 times, with parameters n=1000,k1=400,k2=600,m=100
median of estimates=1.000, error of median estimator=0.000, std= 0.0000025.3
normalized error of median= 0.0001593621193426113 should be <1.0
#### test no. 5
computed prob=0.001, std=0.003
ran estimator 100 times, with parameters n=1000,k1=550,k2=1000,m=100
median of estimates=0.000, error of median estimator=-0.001, std= 0.0027975.3
normalized error of median= 0.27987751426889984 should be <1.0
#### test no. 6
computed prob=0.446, std=0.050
ran estimator 100 times, with parameters n=1000,k1=470,k2=499,m=100
median of estimates=0.440, error of median estimator=-0.006, std= 0.0497065.3
normalized error of median= 0.11861045590584546 should be <1.0