Boolean arrays

# Import the array library
import numpy as np
# Make random number generator.
rng = np.random.default_rng()

We continue to improve on our implementation for solving the three girls problem.

The random choice page gave us a way to simulate the four children of a family in one go:

sexes = [0, 1]
children = rng.choice(sexes, size=4)
children

array([1, 1, 1, 0])

The next step is to get the count of girls from our four simulated children. Because girls are 1 and boys are 0, we can do that with:

# Add up the integers to count the number of girls.
count = np.sum(children)
count

3

Our interest in whether the count value is equal to 3. We can look at that number and write down “Yes” if the number is equal to 3 and “No” otherwise, but we would like the computer to do that routine work for us. We use comparison:

is_three = count == 3
is_three

True

True means our simulation found a family with three girls, and False means we found a family some other number of girls.

In a while, we are going to simulate a very large number of these families, but for now, let us simulate 5 families, in a somewhat laborious way:

# Make an array to store the counts for each family.
counts = np.zeros(5)
# Make five families, store the counts.
family = rng.choice(sexes, size=4)
counts[0] = np.sum(family)
# Second family
family = rng.choice(sexes, size=4)
counts[1] = np.sum(family)
# Third
family = rng.choice(sexes, size=4)
counts[2] = np.sum(family)
# Fourth
family = rng.choice(sexes, size=4)
counts[3] = np.sum(family)
# Fifth.
family = rng.choice(sexes, size=4)
counts[4] = np.sum(family)
# Show the counts
counts

array([2., 2., 3., 1., 3.])

Each value in counts is the number of girls in one simulated family.

Now we have 5 numbers for which we want to ask the question - is this number equal to 3? We would like five corresponding True or False values.

Here is where arrays continue to work their magic - we can get this result with a single expression:

are_three = counts == 3
are_three

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

are_three is an array with 5 elements, one for every element in the array we compared, counts.

are_three is a Boolean array because it contains only Boolean (True, False) values.

We can see what kind of data the array contains by looking at the dtype attribute of the array. Remember, an attribute is a value attached to another value. In this case it is a value attached to the are_three value.

are_three.dtype

dtype('bool')

Each element in are_three has the result of the comparison for the corresponding element. The code above is equivalent to doing:

# Make an array of Boolean type (the "dtype" argument)
are_three_longhand = np.zeros(5, dtype=bool)
# Do the comparisons one by one.
are_three_longhand[0] = counts[0] == 3
are_three_longhand[1] = counts[1] == 3
are_three_longhand[2] = counts[2] == 3
are_three_longhand[3] = counts[3] == 3
are_three_longhand[4] = counts[4] == 3
# Show the result
are_three_longhand

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

Now we want to know how many of the counts values are equal to 3. This is the same as asking how many True values there are in are_three (or are_three_longhand.

We can do this using the np.count_nonzero function. It accepts an array as its argument, and returns the number of non-zero values in the array. It turns out that np.count_nonzero treats True as non-zero, and False as zero, so np.count_nonzero on a Boolean array counts the number of True values:

my_booleans = np.array([True, False, True])
np.count_nonzero(my_booleans)

2

To see the number of times we found 3 in counts:

np.count_nonzero(are_three)

2