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Probability: exercises

Probability: exercises — from Keller & Trotter, Applied Combinatorics.

Practice (5)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Our gang of seven (Alice, Bob, Carlos, Dave, Xing, Yolanda and Zori) are students in a class with a total enrollment of 35. The professor chooses three students at random to go to the board to work challenge problems.

    1. What is the probability that Yolanda is chosen?

    2. What is the probability that Yolanda is chosen and Zori is not?

    3. What is the probability that exactly two members of the club are chosen?

    4. What is the probability that none of the seven members of club are chosen?

  2. Bob says to no one in particular, Did you know that the probability that you will get at least one 7 in three rolls of a pair of dice is slightly less than \(1/2\). On the other hand, the probability that you'll get at least one 5 in six rolls of the dice is just over \(1/2\). Is Bob on target, or out to lunch?

  3. Consider the spinner shown in at the beginning of the chapter.

    1. What is the probability of getting at least one 5 in three spins?

    2. What is the probability of getting at least one 3 in three spins?

    3. If you keep spinning until you get either a 2 or a 5, what is the probability that you get a 2 first?

    4. If you receive \(i\) dollars when the spinner halts in region\(i\), what is the expected value? Since three is right in the middle of the possible outcomes, is it reasonable to pay three dollars to play this game?

  4. Alice proposes to Bob the following game. Bob pays one dollar to play. Fifty balls marked \(1,2,\dots,50\) are placed in a big jar, stirred around, and then drawn out one by one by Zori, who is wearing a blindfold. The result is a random permutation \(\sigma\) of the integers \(1\), \(2,\dots,50\). Bob wins with a payout of two dollars and fifty cents if the permutation \(\sigma\) is a derangement, , \(\sigma(i)\neq i\) for all \(i=1,2,\dots,n\). Is this a fair game for Bob? If not how should the payoff be adjusted to make it fair?

  5. A random graph with vertex set \(\{1,2,\dots,10\}\) is constructed using the following method. For each two element subset \(\{i,j\}\) from \(\{1,2,\dots,10\}\), a fair coin is tossed and the edge \(\{i,j\}\) then belongs to the graph when the result is heads. For each \(3\)-element subset \(S\subseteq\{1,2,\dots,n\}\), let \(E_S\) be the event that \(S\) is a complete subgraph in our random graph.

    1. Explain why \(P(E_S)= 1/8\) for each \(3\)-element subset \(S\).

    2. Explain why \(E_S\) and \(E_T\) are independent when \(|S\cap T|\le 1\).

    3. Let \(S=\{1,2,3\}\), \(T=\{2,3,4\}\) and \(U=\{3,4,5\}\). Show that \[\begin{aligned}\end{aligned}\]

Symbols used here

x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\sigma,\ s,\ \sigma^2
standard deviation, sample s.d., variance
Typical distance from the mean; its square.
P(A),\ P(A \mid B)
probability, conditional probability
Chance of A; chance of A given that B happened.
i
imaginary unit
i² = −1.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\neq
not equal
The two sides are different.
n!
factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
\bar{x},\ \mu
sample mean, population mean
Average of the data; average of the whole population.
E[X],\ \operatorname{Var}(X)
expected value, variance
Probability-weighted average of X; its spread.
N(\mu, \sigma^2),\ z
normal distribution, z-score
The bell curve with mean μ and variance σ²; (x − μ)/σ.
\mu(A),\ \sigma\text{-algebra}
measure of A
Size of a set; the family of sets that can be measured.

Questions people ask

What is the difference between probability and statistics?

Probability goes from a known model to what the data should look like; statistics goes from data back to the model. Probability theory is the deductive half.

What does the law of large numbers promise?

That the average of many independent samples converges to the expected value. It says nothing about any single trial.

Prova själv

Parts of this page are adapted from Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensed and re-explained here; errors are ours.

Mer information Probability Theory

Sample spaces and the axiomsRandom variables and expectationThe common distributionsThe law of large numbers and the central limit theorem