Simulation

Simulation

The production of a computer model of something, especially for the purpose of study.

Simulations are used to estimate probabilities.

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To perform a random experiment on the computer, we use np.random.choice(options).
The input, options, is a list or array to choose from.
The output is a random element in options. By default, all elements are equally likely to be chosen.

# Simulate a fair coin flip.  
np.random.choice(['Heads', 'Tails']) # 'Tails'  
 
# Simulate a roll of a die.  
np.random.choice(np.arange(1, 7)) # '6'  
 
# Simulate 10 fair coin flips.  
np.random.choice(['Heads', 'Tails'], 10)  
# array(['Tails', 'Tails', 'Heads', 'Tails', 'Tails', 'Heads', 'Tails',  
# 'Tails', 'Tails', 'Tails'], dtype='<U5')  

By default, np.random.choice selects with replacement.

# Choose three colleges to win free HDH swag.  
colleges = ['Revelle', 'John Muir', 'Thurgood Marshall',  
'Earl Warren', 'Eleanor Roosevelt', 'Sixth', 'Seventh', 'Eighth']  
 
np.random.choice(colleges, 3, replace=False)  
# array(['Sixth', 'Eighth', 'Seventh'], dtype='<U17')  
np.random.choice(colleges, 3)  
# array(['Sixth', 'Sixth', 'Seventh'], dtype='<U17')  

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What is the probability of getting 60 or more heads if we flip 100 coins?

Plan:

1. Figure out how to run the experiment (flipping 100 coins) once.
2. Repeat the experiment many times.
3. Find the proportion of experiments in which the number of heads was 60 or more.

# Step One  
def coin_experiment():  
	coins = np.random.choice(['Heads', 'Tails'], 100)  
	return np.count_nonzero(coins == 'Heads')  
 
# Step Two  
# Specify the number of repetitions.  
repetitions = 10000  
 
# Create an empty array to store the results.  
head_counts = np.array([])  
 
for i in np.arange(repetitions):  
   # For each repetition, run the experiment and add the result to head_counts.  
   head_count = coin_experiment()  
   head_counts = np.append(head_counts, head_count  
 
# Step Three  
# In how many experiments was the number of heads >= 60?  
at_least_60 = np.count_nonzero(head_counts >= 60)  
at_least_60 # 298  
at_least_60 / repetitions # 0.0298  
np.mean(head_counts >= 60) # 0.0298