SOMETIMES THE QUESTIONS ARE COMPLICATED AND THE ANSWERS ARE SIMPLE
Dr. Seuss
You toss a coin 30 times and see head 24 times. Is it a fair coin?
Hypothesis 1: Tossing a fair coin will get you 15 heads in 30 tosses. This coin is biased
Hypothesis 2: Come on, even a fair coin could show 24 heads in 30 tosses. This is just by chance
P(H) = ?
P(HH) = ?
P(THH) = ?
Now, slightly tougher : P(2H, 1T) = ?
Generalizing,
What is the probability of getting 24 heads in 30 tosses ?
It is the probability of getting heads 24 times or more.
Simulation. Run the experiment 100,000 times. Find the percentage of times the experiment returned 24 or more heads. If it is less than 5%, we conclude that the coin is biased.
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import numpy as np
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total_tosses = 30
num_heads = 24
prob_head = 0.5
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#0 is tail. 1 is heads. Generate one experiment
experiment = np.random.randint(0,2,total_tosses)
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print("Data of the Experiment:", experiment)
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#Find the number of heads
print("Heads in the Experiment:", experiment[experiment==1]) #This will give all the heads in the array
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head_count = experiment[experiment==1].shape[0] #This will get the count of heads in the array
print("Number of heads in the experiment:", head_count)
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#Now, the above experiment needs to be repeated 100 times. Let's write a function and put the above code in a loop
def coin_toss_experiment(times_to_repeat):
head_count = np.empty([times_to_repeat,1], dtype=int)
for times in np.arange(times_to_repeat):
experiment = np.random.randint(0,2,total_tosses)
head_count[times] = experiment[experiment==1].shape[0]
return head_count
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head_count = coin_toss_experiment(100)
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head_count[:10]
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print("Dimensions:", head_count.shape, "\n","Type of object:", type(head_count))
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#Let's plot the above distribution
import matplotlib.pyplot as plt
%matplotlib inline
import seaborn as sns
sns.set(color_codes = True)
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sns.distplot(head_count, kde=False)
Exercise: Try setting kde=True in the above cell and observe what happens
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sns.distplot(head_count, kde=True)
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#Number of times the experiment returned 24 heads.
head_count[head_count>=24]
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print("No of times experiment returned 24 heads or more:", head_count[head_count>=24].shape[0])
print("% of times with 24 or more heads: ", head_count[head_count>=24].shape[0]/float(head_count.shape[0])*100)
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Removing for loop in the funciton
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def coin_toss_experiment_2(times_to_repeat):
head_count = np.empty([times_to_repeat,1], dtype=int)
experiment = np.random.randint(0,2,[times_to_repeat,total_tosses])
return experiment.sum(axis=1)
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