random_number_between_1_to_12

random_number_between_1_to_12

Generate Random numbers between 1-12

Given a function rand6() that returns random numbers from 1 to 6 with equal probability, implement a function rand12() using rand6() that returns random numbers from 1 to 12 with equal probability

Solution

Call rand6() twice and generate two numbers , let us call them $n u m_{1}$ and $n u m_{2}$
$n u m_{1} = r a n d 6 ()$
$n u m_{2} = r a n d 6 ()$

Generate the output number based on the below formula
$\begin{array}{r} o u t p u t = {\begin{array}{cl} n u m_{1} + 6 & (i f n u m_{2} i s e v e n) \\ n u m_{1} & o t h e r w i s e . \end{array} \end{array}$

Explanation of the above formula

If above formula is correct then probability of getting any number between 1-12 should be uniform i;e, p = 1/12.

Let's Estimate Probabilities

Case 1 : If num_2 is odd, then num_1 can be anything $[1 - 6]$ . In this case output number will be $[1, 6]$ with $\begin{array}{r} P (n u m_{1} = [1 - 6]) * P (n u m_{2} i s o d d) = \frac{1}{6} * \frac{3}{6} = \frac{1}{12} \end{array}$

Case 2 : if num_2 is even, then num_1 can be anything $[1, 6]$ . In this case output number will be $[7, 12]$ with $\begin{array}{r} P (n u m_{1} = [1 - 6]) * P (n u m_{2} i s e v e n) = \frac{1}{6} * \frac{3}{6} = \frac{1}{12} \end{array}$


import numpy as np

# Function that returns random numbers from 1 to 6 with equal probability
def rand6():
    return np.random.randint(1, 6)

# The function uses rand6() to return random numbers from 1 to 12 with equal probability
def rand12():
    num = None
    # Write your code
    num = rand6() + (rand6()%2 ) *6
    return num