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Homework Week 14

Discrete Mathematics · 16 problems · solutions not included

Flashcards

Problems

Section 7.3 - Bayes’ Theorem

Problem 2

Given:

\[ P(E)=\frac23,\quad P(F)=\frac34,\quad P(F\mid E)=\frac58. \]

Find $P(E\mid F)$.

Section 7.3 - Bayes’ Theorem

Problem 4

Ann chooses one of two boxes at random, then chooses one ball from that box.

Box 1: 3 orange balls, 4 black balls.
Box 2: 5 orange balls, 6 black balls.

Find $P(\text{Box 2}\mid \text{orange ball selected})$.

Section 7.3 - Bayes’ Theorem

Problem 6

A steroid test is given to soccer players.

Given: 5% of soccer players use steroids; 98% of steroid users test positive; 12% of non-users test positive.

Find $P(\text{player uses steroids}\mid \text{player tests positive})$.

Section 7.3 - Bayes’ Theorem

Problem 8

A rare genetic disease occurs in 1 out of every 10,000 people. The test works as follows: 99.9% of people with the disease test positive, and 0.02% of people without the disease test positive.

Find $P(\text{has disease}\mid \text{tests positive})$.
Find $P(\text{does not have disease}\mid \text{tests negative})$.

Section 7.3 - Bayes’ Theorem

Problem 9

In a clinic, 8% of patients are infected with HIV. The blood test works as follows: 98% of infected patients test positive, and 3% of non-infected patients test positive.

$P(\text{infected}\mid \text{tests positive})$
$P(\text{not infected}\mid \text{tests positive})$
$P(\text{infected}\mid \text{tests negative})$
$P(\text{not infected}\mid \text{tests negative})$

Section 7.3 - Bayes’ Theorem

Problem 11

A company is introducing a new camera phone. Past data: 60% of new products are successful, 70% of successful products were predicted to succeed, and 40% of failed products were predicted to succeed.

Find $P(\text{product is successful}\mid \text{product was predicted to succeed})$.

Section 7.3 - Bayes’ Theorem

Problem 13

Events $F_1,F_2,F_3$ are pairwise disjoint and together form the whole sample space $S$.

Given:

\[ P(E\mid F_1)=\frac18, \quad P(E\mid F_2)=\frac14, \quad P(E\mid F_3)=\frac16, \]

\[ P(F_1)=\frac14, \quad P(F_2)=\frac14, \quad P(F_3)=\frac12. \]

Find $P(F_1\mid E)$.

Section 7.3 - Bayes’ Theorem

Problem 15

Monty Hall problem. There are 3 doors. One door has the prize. The other two doors are losing doors. You choose one door. Monty opens a losing door that you did not choose. If Monty has a choice between two losing doors, he chooses randomly. Monty then asks whether you want to switch.

Let $W$ be the winning door and $M$ the door Monty opens. Assume $P(W=k)=1/3$ for $k=1,2,3$. Suppose you initially choose door $i$.

Find the probability of winning if you stay with your original door.
Find $P(M=j\mid W=k)$ for $j=1,2,3$ and $k=1,2,3$.
Use Bayes’ theorem to find $P(W=j\mid M=k)$ when $i,j,k$ are distinct.
Explain whether the answer to part (c) tells you to switch doors.

Section 7.4 - Expected Value and Variance

Problem 5

Two biased dice are rolled. Each die is biased so that 3 is twice as likely to appear as any other number. Find the expected value of the sum of the two dice.

Section 7.4 - Expected Value and Variance

Problem 6

A $1 lottery ticket is bought. The lottery chooses 6 winning numbers from $\{1,2,3,\dots,50\}$. The purchaser wins $10,000,000 if their ticket contains the 6 winning numbers. Otherwise, the purchaser wins nothing. Find the expected value of buying the ticket.