5.1.5 Practice set: probability

Sample spaces, events and axioms

Sample spaces, events and the axioms (lesson 5.1.1)

0 of 6 correct

1.
A die is rolled and a coin is tossed. How many outcomes does the sample space have?
2.
One die: $A = \{1, 2, 3\}$ and $B = \{2, 4, 6\}$ . How many outcomes are in $A \cap B^c$ ?
3.
$A$ and $B$ are disjoint, with $P(A) = 0.45$ and $P(B) = 0.3$ . Find $P((A \cup B)^c)$
4.
Two fair dice are rolled. Find the probability that the sum is at most 4
5.
$P(A) = 0.7$ and $P(B) = 0.6$ . What is the smallest that $P(A \cap B)$ can be?
6.
Two fair dice are rolled. Find the probability that one shows 1 and the other shows 2

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Counting

Counting (lesson 5.1.2)

0 of 6 correct

1.
In how many orders can the 5 letters of MATHS be written?
2.
How many 4-digit PINs have all four digits different?
3.
A pizza takes 3 different toppings from a list of 9. How many choices of toppings are there?
4.
A 5-card hand is dealt. Find the probability that it has no hearts
5.
A committee of 4 is picked at random from 5 juniors and 3 seniors. Find the probability that it has exactly 2 seniors
6.
How many 5-card hands contain at least one ace?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Conditional probability and independence

Conditional probability and independence (lesson 5.1.3)

0 of 7 correct

1.
$P(A) = 0.5$ , $P(B) = 0.4$ and $P(A \cap B) = 0.1$ . Find $P(B \mid A)$
2.
An urn has 4 red and 6 blue balls. Two are drawn without replacement. Find the probability that both are blue
3.
30% of days are rainy. A train is late with probability 0.4 on a rainy day and 0.1 otherwise. Find the probability that it is late
4.
A fair coin is tossed 3 times. Find the probability of exactly one head
5.
$A$ and $B$ are independent, with $P(A) = 0.4$ and $P(B) = 0.5$ . Find $P(A \cap B^c)$
6.
Two fair dice: given that at least one shows a 6, find the probability that both do
7.
$A$ and $B$ are mutually exclusive, with $P(A) = 0.3$ and $P(B) = 0.4$ . Find $P(A \cap B)$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Bayes' rule

Bayes' rule (lesson 5.1.4)

0 of 6 correct

1.
$P(A) = 0.3$ , $P(B \mid A) = 0.5$ and $P(B \mid A^c) = 0.2$ . Find $P(B)$
2.
With the same numbers, find $P(A \mid B)$
3.
Urn 1 holds 3 red and 1 blue ball; urn 2 holds 1 red and 3 blue. An urn is picked at random and a red ball drawn. Find the probability that it was urn 1
4.
A condition has a base rate of 0.5%. A test has a 99% hit rate and a 2% false-alarm rate. Find $P(\text{has it} \mid +)$
5.
The prior odds are 1 to 9 and the likelihood ratio is 6. Find the posterior probability
6.
The prior is 0.2. A test with a 90% hit rate and a 10% false-alarm rate comes back negative. Find the probability of the condition

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

Exactly k special items in a draw

Lesson 5.1.2 found the chance of exactly $k$ special items when $n$ items are drawn from $N$ , of which $K$ are special: $\binom{K}{k}\binom{N - K}{n - k}$ over $\binom{N}{n}$ . Write two functions.

draw_exactly(N, K, n, k) returns that probability as a float. If $k$ cannot happen, for example more aces than cards dealt, return 0.0. Note that math.comb raises ValueError for a negative argument, so check the range first.
draw_at_least(N, K, n, k) returns the chance of $k$ or more special items. It is 1 when $k \le 0$ .

Both raise ValueError if $N$ , $K$ or $n$ is negative, or if $K > N$ or $n > N$ .

The tests use two aces in a hand (0.0399) and the lottery (0.01765 for exactly 3 matches, 0.01864 for at least 3). They also try impossible draws, and check that the probabilities for $k = 0, \ldots, n$ add up to 1.

See the tests

def _raises(f, *args, what=""):
    try:
        f(*args)
    except ValueError:
        return
    raise AssertionError(f"expected ValueError for {what}")


def test_two_aces():
    """Exactly two aces in a 5-card hand has probability about 0.0399"""
    expect_close(draw_exactly(52, 4, 5, 2), 0.03992981808107859, what="draw_exactly(52, 4, 5, 2)", rel=1e-9)


def test_lottery():
    """Matching exactly 3 of 6 numbers drawn from 49 has probability about 0.01765"""
    expect_close(draw_exactly(49, 6, 6, 3), 0.017650403866870102, what="draw_exactly(49, 6, 6, 3)", rel=1e-9)


def test_all_hearts():
    """Five hearts in a 5-card hand has probability about 0.000495"""
    expect_close(draw_exactly(52, 13, 5, 5), 0.0004951980792316927, what="draw_exactly(52, 13, 5, 5)", rel=1e-9)


def test_impossible_draws():
    """Impossible counts give 0: five aces, 3 aces in 2 cards, too few others, or k = -1"""
    expect_close(draw_exactly(52, 4, 5, 5), 0.0, what="five aces in a hand")
    expect_close(draw_exactly(52, 4, 2, 3), 0.0, what="three aces in two cards")
    expect_close(draw_exactly(10, 8, 5, 2), 0.0, what="2 special of 5 when only 2 others exist")
    expect_close(draw_exactly(52, 4, 5, -1), 0.0, what="k = -1")


def test_adds_up_to_one():
    """The chances of k = 0, 1, ..., n special items add up to 1"""
    total = sum(draw_exactly(49, 6, 6, k) for k in range(7))
    expect_close(total, 1.0, what="sum over k for the lottery", rel=1e-12)
    total = sum(draw_exactly(10, 6, 3, k) for k in range(4))
    expect_close(total, 1.0, what="sum over k for a committee of 3 from 6 juniors and 4 seniors", rel=1e-12)


def test_at_least():
    """At least 3 lottery matches is about 0.01864; at least one ace in a hand about 0.3412"""
    expect_close(draw_at_least(49, 6, 6, 3), 0.01863754500202234, what="draw_at_least(49, 6, 6, 3)", rel=1e-9)
    expect_close(draw_at_least(52, 4, 5, 1), 0.34115800166220334, what="draw_at_least(52, 4, 5, 1)", rel=1e-9)


def test_at_least_edges():
    """At least 0 is certain, and at least 5 aces in a hand is impossible"""
    expect_close(draw_at_least(52, 4, 5, 0), 1.0, what="draw_at_least(52, 4, 5, 0)")
    expect_close(draw_at_least(52, 4, 5, 5), 0.0, what="draw_at_least(52, 4, 5, 5)")


def test_bad_inputs():
    """Drawing more than N, more special items than N, or a negative size raises ValueError"""
    _raises(draw_exactly, 10, 4, 11, 2, what="n > N")
    _raises(draw_exactly, 10, 12, 3, 2, what="K > N")
    _raises(draw_exactly, 10, -1, 3, 0, what="K < 0")
    _raises(draw_at_least, 10, 4, 11, 2, what="n > N in draw_at_least")

Sample spaces, events and axioms

Counting

Conditional probability and independence

Bayes' rule

Coding exercise

Free further reading