# Robert Coding Interview

Simulate a coin flip using nothing but the standard Python library.

Things to look out for:

1. Docstrings
2. Functional programming
3. [DONE] Sanity-checking implementation (p vs. 1-p)
4. Defensive programming checks for input

#input n = number of coin flips
#output = number of heads
import random

# functional style
# multiply all numbers from 1 to 10
mul(range(1, 11))

# vs. imperative
res = 1
for i in range(1, 11):
    res *= i
    

ST  -> H   -> T  => 1.1 (final result)
ST  -> T   -> H  => 1.1 (final result)
1.0 -> 1.1 -> 1.1 -> 

H H T T H -> [('H', 2), ('T', 2), ('H', 1)]


def coin_flip(n: int, p: float):
    '''
    Return the number of heads from n coin flips.
    
    :param n: Number of coin flips to simulate.
    :param p: Probability for heads coin flip.
    :returns: Number of heads.
    '''
    # Defensive programming
    if (p > 1) or (p < 0):
        # ValueError is the right thing - this is a builtin to the Python language
        raise ValueError("p should be between 0 and 1 inclusive.")
    
    sequence = []
    heads = 0
    for i in range(n):
        rand = random()
        if rand > 1-p:
            heads += 1
            
            if i == 0:
                sequence.append(['H', 1])
            elif sequence[-1][0] == 'H':
                sequence[-1][1] += 1
            else:
                sequence.append(['T', 1])
        else:
            if i == 0:
                sequence.append(['T', 1])
            elif sequence[-1][0] == 'T':
                sequence[-1][1] += 1
            else:
                sequence.append(['H', 1])
                
    return {sequence, heads}
    # ans = sum([random() > 0.5 for _ in range(n)])
    #return sum(map(lambda : random() > 1-p, range(n)))



def coin_flips(n: int):
    '''
    Return the number of heads from n coin flips.
    
    :param n: Number of coin flips to simulate.
    :returns: Number of heads.
    '''
    ans_sum = 0
    
    for i in range(n):
        rand = random() # 0-1
        
        if rand > 0.5:
            ans_sum += 1
            
    return ans_sum

You're N coin flips with H heads.
What is the estimated value of `p`,
and its uncertainty bounds?

#input H heads, n flips
#output p and sigma(p)

# prior Beta(1, 1)  beta-distributed with equal weight on "H" and "T".
# likelihood Binom(n trials, H success)
# # because Beta distribution is a conjucate prior for Binomial likelihoods,
# # the posterior can be analytically calculated as a Beta distribution.
# posterior Beta(1+H, 1+n-H)
# to calculate uncertainty, take % bounds on the CDF of posterior distribution
from scipy.stats import beta

def estimate_prob(H: int, n: int):
    '''
    Determine probablity of heads and uncertain of probability.
    
    # Assumes prior distribution of Beta(1, 1).
    
    :param H: number of heads
    :param n: number of coin flip
    '''
    if H > n:
        raise ValueError('Number of heads must be less than number of flips')
    if (H < 0) or (n < 0):
        raise ValueError('Number of heads and number of flips must be positive')
        
    posterior = beta(1 + H, 1 + n - H)
    expectation = posterior.cdf(0.5)
    lowerbound = posterior.cdf(0.05)
    upperbound = posterior.cdf(0.95)
    
    return expectation, (lowerbound, upperbound)
        
    
    p = float(H)/n
    #delta_p = n*p*(1-p)
    delta_p_plus = 
    delta_p_minus = 

    return p, delta_p