Three girl simulations

Exercises on the three-girl simulation problem

These exercises follow on from the three girls solution in filling arrays, and the reply to the Supreme Court.

You may want to refer back to those pages for inspiration.

Run this cell first:

# Load the Numpy package, and rename to "np"
import numpy as np

Three girls in a family of four

Remember the solution to this problem, from filling arrays:

# Reset the counts array to empty (all zeros)
counts = np.zeros(10000)

# Repeat the indented stuff 10000 times.
for i in np.arange(10000):
    # The procedure for one family.
    family = np.random.randint(0, 2, size=4)
    count = np.sum(family)
    # Fill in the corresponding value in the counts array.
    counts[i] = count

# Proportion
prop_3_girls = np.count_nonzero(counts == 3) / 10000
prop_3_girls

0.2453

No girls in a family of four

Estimate the chances that a 4-child family will have no girls. You can copy the code from the cell above, and modify it, or you may be able to use variables from the code above, to get the answer, without repeating the simulation.

# Show proportion with 0 girls.
# Your code below

For extra points - the answer above is easier to work out with probability than the chance of three girls. What’s the exact answer, from probability?

Edit this code cell to give your answer.

Three girls in a family of five

Simulate the chances that a family with 5 children will have exactly 3 girls.

# Simulate 10000 families of 5 children.
# Show proportion with 3 girls.
# Your code below

Three or fewer girls in a family of five

Simulate the chances that a family with 5 children will have 3 or fewer girls.

Hint: you may remember from the Comparison page that <= tests whether the thing on the left is less than or equal to the thing on the right.

3 <= 4

True

3 <= 3

True

3 <= 2

False

my_array = np.array([1, 2, 3, 4])
my_array <= 2

array([ True,  True, False, False])

Hints done, now:

# Simulate 10000 families of 5 children.
# Show the proportion of families of 5 children with 3 or fewer girls.
# Your code below

More realistic simulation

Now we are back to the situation of exactly 3 girls in a family of 4.

In fact, when you have a child, the probability of having a girl is slightly less than 0.5.

In fact, the proportion of boys born in the UK is 0.513. Hence the proportion of girls is 1-0.513 = 0.487.

With that probability of having a girl, what are the chances of having exactly three girls in a family of four?

Hint 1: you may want to use np.random.uniform. Check the help with np.random.uniform? followed by Shift-Enter in a new cell. It works like this:

# An array of four random numbers between 0 and 1.
np.random.uniform(0, 1, size=4)

array([0.11587267, 0.16510294, 0.41035455, 0.21159489])

Hint 2: Let’s say I have a random number x between 0 and 1:

x = np.random.uniform(0, 1)
x

0.7525928240586782

The probability that that x will be less than - say - 0.25, is 0.25.

# Simulate 10000 familes of 4 children.
chance_of_girl = 0.487
# Estimate chance of having exactly 3 girls.
# Your code here