Description
problem 1 (Census).
Random Census, Inc. — a data science company — conducted a study of families
that have 1, 2, or 3 children, all having different ages (i.e., no family surveyed had twins). The data
scientists randomly sampled families and, for each sampled family, they wrote down the genders of the
children in ascending order.
For example, the outcome G indicates that the family has one child, who
is a girl; the outcome GB indicates that the family has two children, the younger child is a girl, and the
older child is a boy. They performed a statistical analysis of the data and concluded that every possible
configuration of genders and numbers of children is equally likely.
(a) Write down the sample space of this experiment and the probability of each outcome.
(b) For each of the following events, write down the set of outcomes comprising the event and find the
probability of the event.
A = “families where the oldest child is a boy”
B = “families with exactly one girl and any number of boys”
C = “families with at least one child of each gender”
Solution: Your solution here.
problem 2 (Random passwords).
The passwords for a certain website are sequences comprised of 4 numbers and 5 letters, where each number is one of the 10 digits {0, 1, 2, . . . , 9} and each letter is one of the
upper-case letters of the English alphabet {A, B, . . . , Z}. Suppose we choose a password uniformly at
random (each password is equally likely to be chosen). Find the probability of the event that the letters
come before the numbers.
Solution: Your solution here.
problem 3 (Rolling dice).
We have a fair 4-sided die with the numbers 1, 2, 3, 4 written on its faces. Suppose
we roll the die n times. Each outcome is equally likely. Find the probability of getting:
(a) All threes.
(b) No threes.
(c) Not all threes.
(d) At least one three.
Solution: Your solution here.
problem 4 (Combination of events).
Let A, B, C be three events. Using set notation and operations, write
down each of the following combinations of the events:
(a) A and C occur, but not B.
(b) One of B or C occurs, bot not both of them.
(c) Neither of the events occurs.
(d) At least one of the events occurs.
(e) All of the events occur.
(f) Only one of the events occurs.
Solution: Your solution here.
problem 5 (Pokemon Go).
Alice and Bob take part in a special Pokemon Go event where each of them
gets to win one of three prizes: a Pikachu, a Sinoh Stone, or a Technical Machine. The pair of prizes are
randomly selected (not necessarily uniformly or independently of each other).
(a) Write a suitable sample space for this experiment.
(b) Suppose that:
– the probability that Alice gets a Pikachu is 0.2;
– the probability that Bob gets a Pikachu is 0.5;
– the probability that at least one gets a Pikachu but neither gets a Sinoh Stone is 0.3.
Find the probability that at least one gets a Sinoh Stone but neither gets a Technical Machine.
Solution: Your solution here.
problem 6 (Programming exercises).
Download this Jupyter notebook. Complete all the exercises in the
notebook. Submit the Jupyter notebook with your solutions to the Homework 2 Programming assignment
on Gradescope. Your submission should be a single .ipynb file.
H2-2
