Solutions#

Question 1#

1. For each of the following, Write a function, and repeatedly use it to simulate the probability of an event occurring with the following chances:

1. \(\frac{2}{7}\)

import random


def sample_experiment():
    """
    Returns true if a random number is less than 2 / 7
    """
    return random.random() < 2 / 7


number_of_experiments = 1000
sum(
    sample_experiment() for repetition in range(number_of_experiments)
) / number_of_experiments

0.284

2. \(\frac{1}{10}\)

def sample_experiment():
    """
    Returns true if a random number is less than 1 / 10
    """
    return random.random() < 1 / 10


number_of_experiments = 1000
sum(
    sample_experiment() for repetition in range(number_of_experiments)
) / number_of_experiments

0.113

3. \(\frac{1}{100}\)

def sample_experiment():
    """
    Returns true if a random number is less than 1 / 100
    """
    return random.random() < 1 / 100


number_of_experiments = 1000
sum(
    sample_experiment() for repetition in range(number_of_experiments)
) / number_of_experiments

0.011

4. \(1\)

def sample_experiment():
    """
    Returns true if a random number is less than 1
    """
    return random.random() < 1


number_of_experiments = 1000
sum(
    sample_experiment() for repetition in range(number_of_experiments)
) / number_of_experiments

1.0

Attention

We can in fact write a single function for all the probabilities here:

def sample_experiment(probability):
    """
    Returns true of a random number is less than probability
    """
    return random.random() < probability

Question 2#

2. Write a function, and repeatedly use it to simulate the probability of selecting a red token from each of the following configurations:

1. A bag with 4 red tokens and 4 green tokens.

def sample_experiment(number_of_red, number_of_green, number_of_yellow):
    """
    This samples a token from a bag with number_of_red red tokens,
    number_of_green green tokens and number_of_yellow yellow tokens
    """
    bag = (
        ["Red"] * number_of_red
        + ["Green"] * number_of_green
        + ["Yellow"] * number_of_yellow
    )

    return random.choice(bag)

We now use this function:

number_of_experiments = 1000
random.seed(0)
sum(
    sample_experiment(number_of_red=4, number_of_green=4, number_of_yellow=0) == "Red"
    for repetition in range(number_of_experiments)
) / number_of_experiments

0.503

2. A bag with 4 red tokens, 4 green tokens and 10 yellow tokens.

random.seed(0)
sum(
    sample_experiment(number_of_red=4, number_of_green=4, number_of_yellow=10) == "Red"
    for repetition in range(number_of_experiments)
) / number_of_experiments

0.241

3. A bag with 0 red tokens, 4 green tokens and 10 yellow tokens.

random.seed(0)
sum(
    sample_experiment(number_of_red=0, number_of_green=4, number_of_yellow=10) == "Red"
    for repetition in range(number_of_experiments)
) / number_of_experiments

0.0

Question 3#

3. An experiment consists of selecting a token from a bag and spinning a coin. The bag contains 3 red tokens and 4 blue tokens. A token is selected at random from the bag, its colour is noted and then the token is returned to the bag. When a red token is selected, a biased coin with probability \(\frac{4}{5}\) of landing heads is spun. When a blue token is selected, a biased coin with probability \(\frac{2}{5}\) of landing heads is spun.

1. Approximate the probability of picking a red token?

We will use the code from the tutorial, making the necessary modifications:

def pick_a_token(container):
    """
    A function to randomly sample from a `container`.
    """
    return random.choice(container)


def sample_experiment(bag):
    """
    This samples a token from a given bag and then
    selects a coin with a given probability.

    If the sampled token is red then the probability
    of selecting heads is 4/5 otherwise it is 2/5.

    This function returns both the selected token
    and the coin face.
    """
    selected_token = pick_a_token(container=bag)

    if selected_token == "Red":
        probability_of_selecting_heads = 4 / 5
    else:
        probability_of_selecting_heads = 2 / 5

    if random.random() < probability_of_selecting_heads:
        coin = "Heads"
    else:
        coin = "Tails"
    return selected_token, coin

bag = ["Red"] * 3 + ["Blue"] * 4
number_of_repetitions = 10000
random.seed(0)
samples = [sample_experiment(bag=bag) for repetition in range(number_of_repetitions)]
sum(token == "Red" for token, coin in samples) / number_of_repetitions

0.4313

2. Approximate the probability of obtaining Heads?

sum(coin == "Heads" for token, coin in samples) / number_of_repetitions

0.5727

3. If a heads is obtained, approximate the probability of having selected a red token.

samples_with_heads = [(token, coin) for token, coin in samples if coin == "Heads"]
sum(token == "Red" for token, coin in samples_with_heads) / len(samples_with_heads)

0.5978697398288807

Question 4#

4. On a randomly chose day, the probability of an individual travelling to school by car, bicycle or on foot is \(1/2\), \(1/6\) and \(1/3\) respectively. The probability of being late when using these methods of travel is \(1/5\), \(2/5\) and \(1/10\) respectively.

1. Approximate the probability that an individual travels by foot and is late.

We will write a function to sample the experiment.

def sample_experiment():
    """
    Sample a travel method followed by whether or not the individual is late.
    """
    random_number = random.random()
    if random_number < 1 / 2:
        travel_method = "car"
        is_late = random.random() < 1 / 5
        return travel_method, is_late
    if random_number < 1 / 2 + 1 / 6:
        travel_method = "bicycle"
        is_late = random.random() < 2 / 5
        return travel_method, is_late
    travel_method = "foot"
    is_late = random.random() < 1 / 10
    return travel_method, is_late

We will use this to now obtain a number of samples:

number_of_repetitions = 10000
random.seed(0)
samples = [sample_experiment() for repetition in range(number_of_repetitions)]
samples

[('foot', False),
 ('car', False),
 ('bicycle', False),
 ('foot', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('bicycle', True),
 ('foot', False),
 ('foot', False),
 ('car', False),
 ('foot', False),
 ('car', True),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('bicycle', True),
 ('foot', False),
 ('foot', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('bicycle', False),
 ('car', False),
 ('foot', False),
 ('foot', False),
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 ('bicycle', False),
 ('bicycle', True),
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 ('car', True),
 ('bicycle', False),
 ('car', True),
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 ('bicycle', True),
 ('foot', False),
 ('foot', False),
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 ('car', False),
 ('car', False),
 ('foot', False),
 ('foot', False),
 ('car', False),
 ('car', False),
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 ('foot', True),
 ('bicycle', False),
 ('bicycle', True),
 ('car', False),
 ('car', True),
 ('foot', False),
 ('car', False),
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 ('foot', False),
 ('car', False),
 ('foot', False),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('bicycle', True),
 ('car', False),
 ('car', False),
 ('car', False),
 ('bicycle', True),
 ('foot', False),
 ('car', True),
 ('bicycle', False),
 ('car', False),
 ('car', False),
 ('foot', True),
 ('car', False),
 ('foot', False),
 ('foot', False),
 ('car', True),
 ('car', False),
 ('car', False),
 ('foot', False),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('car', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('bicycle', False),
 ('bicycle', False),
 ('bicycle', False),
 ('car', False),
 ('foot', False),
 ('foot', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('foot', False),
 ('foot', False),
 ('car', False),
 ('foot', True),
 ('foot', False),
 ('foot', False),
 ('car', False),
 ('car', True),
 ('foot', False),
 ('bicycle', True),
 ('car', False),
 ('bicycle', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('foot', True),
 ('foot', False),
 ('car', False),
 ('bicycle', True),
 ('bicycle', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('foot', False),
 ('bicycle', True),
 ('car', False),
 ('car', False),
 ('car', False),
 ('bicycle', True),
 ('foot', False),
 ('car', False),
 ('foot', False),
 ('foot', True),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('car', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('car', True),
 ('bicycle', True),
 ('car', False),
 ('foot', False),
 ('foot', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('foot', False),
 ('foot', False),
 ('foot', False),
 ('foot', False),
 ('car', True),
 ('bicycle', True),
 ('bicycle', False),
 ('car', True),
 ('car', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('bicycle', True),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('bicycle', True),
 ('car', False),
 ('car', False),
 ('bicycle', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('foot', False),
 ('foot', False),
 ('foot', False),
 ('car', False),
 ('bicycle', False),
 ('bicycle', False),
 ('foot', False),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('car', False),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('foot', True),
 ('foot', False),
 ('car', True),
 ('foot', False),
 ('foot', True),
 ('foot', True),
 ('bicycle', False),
 ('foot', False),
 ('foot', False),
 ('car', False),
 ('car', False),
 ('foot', False),
 ('foot', True),
 ('bicycle', True),
 ...]

Using this we can approximate the probability:

sum(
    is_late and travel_method == "foot" for travel_method, is_late in samples
) / number_of_repetitions

0.0351

2. Approximate the probability that an individual is not late.

sum(not is_late for travel_method, is_late in samples) / number_of_repetitions

0.7989

3. Given that an individual is late, approximate the probability that they did not travel on foot.

samples_with_late = [
    (travel_method, is_late) for travel_method, is_late in samples if is_late is True
]
number_of_samples = len(samples_with_late)
sum(
    travel_method != "foot" for travel_method, is_late in samples_with_late
) / number_of_samples

0.8254599701640974

Python for mathematics

Solutions

Contents

Solutions#

Question 1#

Question 2#

Question 3#

Question 4#