# Category Archives: Independence

## Boxes with Yellow and Green Balls

Three boxes contain yellow and green balls

• Box 1 contains 2 yellow balls.
• Box 2 contains 2 green balls.
• Box 3 contains 1 yellow ball and 1 green ball.

One box is selected at random, and one ball is pulled out of that box.

1.  The ball that is pulled out of the chosen box is yellow. What is the probability that the other ball in that same box is also yellow?
2. Let $$A$$ be the event that Box 3 is chosen. Let $$B$$ be the event that a yellow ball is pulled out of the chosen box. Are $$A$$ and $$B$$ independent?
3.  The ball that is pulled out of the chosen box is yellow. Without replacement, a second ball is chosen at random from one of the three boxes. (Each box has a 1/3 chance of being selected.) What is the probability that the second ball chosen is also yellow?

## Flipping Coins and Independence

An experimenter has two fair coins and one biased coin. The biased coin lands on heads with probability 3/4.

The experimenter randomly selects one of the three coins and flips it until they get heads.

Let $$A$$ be the event that the experimenter flipped the biased coin.
Let $$B$$ be the event that it took the experimenter an even number of flips to get heads.

Are events $$A$$ and $$B$$ independent?

## Coin flipping game

Your friend challenges you to a game in which you flip a fair coin until you get heads. If you flip an even number of times, you win. Let $$A$$ be the event that you win. Let $$B$$ be the event that you flip the coin 3 or more times. Let $$C$$ be the event that you flip the coin 4 or more times.

1. Compute $$\mathbb{P}(A)$$.
2. Are $$A$$ and $$B$$ independent?
3. Are $$A$$ and $$C$$ independent?

## Dice Rolling Events

Consider rolling a fair 6-sided die twice. Let $$A$$ be the event that the first roll is less than or equal to 3. Let $$B$$ be the event that the second roll is less than or equal to 3. Find an event $$C$$ in the same outcome space as $$A$$  and $$B$$ with $$0<\mathbb{P}(C)<1$$ and  such that $$A$$, $$B$$ and $$C$$ are mutually independent, or show that no such event exists.

## Three Random Variables

Let $$X$$, $$Y$$, and $$Z$$ be independent uniform $$(0,1)$$.

1. Find the joint density of  $$XY$$ and $$Z^2$$.
2. Show that $$P(XY < Z^2) = \frac59$$.

[Meester ex. 5.12.25]

## A joint density example I

Let $$(X,Y)$$ have joint density $$f(x,y)=x e^{-x-y}$$ when $$x,y>0$$ and $$f(x,y)=0$$ elsewhere. Are $$X$$ and $$Y$$ independent ?

[Meester ex 5.12.30]

## Prime Dice

Suppose that we have a very special die which has exactly $$k$$ faces where $$k$$ is prime. The faces are numbered $$1,\dots,k$$. We throw the die once and see which number comes up.

1. What would be an appropriate outcome space and probability measure for this random experiment ?
2. Suppose that the events $$A$$ and $$B$$ are independent. Show that $$\mathbf{P}(A)$$ or $$\mathbf{P}(B)$$ is always either 0 or 1. Or in other wards $$A$$ or $$B$$ is always either the full space or the empty set.

[ from Meester, ex 1.7.32]

## Independence of two hearts ?

Consider a deck of  52 cards.  Let $$A$$ be the event that the first card is a heart.  Let $$B$$ be the event that the 51st card is a heart.

What is $$\mathbf{P}(A)$$ ? What is $$\mathbf{P}(B)$$ ? Are $$A$$ and $$B$$ independent ?

## Coin tosses: independence and sums

A fair coin is tossed three times. Let $$X$$ be the number of heads on the first two tosses, $$Y$$ the number of heads on the last two tosses.

1. Make a table showing the joint distribution of $$X$$ and $$Y$$.
2. Are $$X$$ and $$Y$$  independent ?
3. Find the distribution of $$X+Y$$ ?

## Algebras and Conditioning

Consider two draws from a box with replacement contain 1 red ball and 3 blue balls. Let $$X$$ be number of red balls. Let $$Y$$ be 1 if the two balls are the same color and 0 otherwise. Let $$Z_i$$ be the random variable which returns 1 if the $$i$$-th ball is red.

1. What is the sample space.
2. Write down the algebra of all events on this sample space.
3. What is the algebra of events generated by $$X$$ ?
4. What is the algebra of events generated by $$Y$$ ?
5. What is the algebra of events generated by $$Z_1$$ ?
6. What is the algebra of events generated by $$Z_2$$ ?
7. Which random variables are determined by an another of  the random variables. Why ? How is this reflected in the algebras ?
8. (*) What pair of random variables are independent ? How is this reflected in the algebras ?

## Only Pairwise Independence

Let $$X_1$$ and $$X_2$$ be two independent tosses of a fair coin. Let $$Y$$ be the  random variable  equal to 1 if exactly one of those coin tosses resulted in heads, and 0 otherwise. For simplicity, let 1 denote heads and 0 tails.

1. Write down the joint probability mass function for $$(X_1,X_2,Y)$$.
2. Show that $$X_1$$ and $$X_2$$ are independent.
3. Show that $$X_1$$ and $$Y$$ are independent.
4. Show that $$X_2$$ and $$Y$$ are independent.
5. The three variables are mutually independent if  for any $$x_1 \in Range(X_1), x_2\in Range(X_2), y \in Range(Y)$$ one has
$\mathbf{P}(X_1=x_1,X_2=x_2,Y=y) = \mathbf{P}(X_1=x_1) \mathbf{P}(X_2=x_2) \mathbf{P}(Y=y)$
Show that $$X_1,X_2,Y$$ are not mutually independent.

## Chance of an Accident.

An insurance company has 50% urban and 50% rural customers. If every year each urban customer has an accident with probability $$\mu$$ and each rural customer has an accident with probability $$\lambda$$. Assume that the chance of an accident is independent from year to year and from customer to costumer. This is another way to say, conditioned on being and urban or rural the chance of having an accident each year is independent.

A costumer is randomly chosen. Let $$A_n$$ be the chance this customer has an accident in year $$n$$.  Let $$U$$ denote the event that this costumer is urban and $$R$$ the event that the customer is rural.

1. Find $$\mathbf{P}(A_2|A_1)$$.
2. Are $$A_1$$ and $$A_2$$ independent in general ? Are there any conditions when it is true  if not in general ?
3. Show that $$\mathbf{P}(A_2|A_1) \geq \mathbf{P}(A_2)$$.
To answer this question it is useful to know that for any positive $$a$$ and $$b$$, one has $$(a+b)^2 < 2(a^2 +b^2)$$ as long as $$a \neq b$$. In the case $$a = b$$, one has of course $$(a+b)^2 = 2(a^2 +b^2)$$. To prove this inequality, first show that $$(a+b)^2 +(a-b)^2= 2(a^2 +b^2)$$ and then use that fact that $$(a-b)^2 >0$$.
4. Find the probability that a driver  has an accident in the 3nd year given that they had one in the 1st and 2nd year.
5. Find the probability that a driver  has an accident in the $$n$$-th year given that they had one in all of the previous years. What is the limit as $$n \rightarrow \infty$$ ?
6. Find the probability that a diver is a urban diver given that they had an accident in two successive years.

## Blocks of Bernoulli Trials

In $$n+m$$ independent  Bernoulli $$(p)$$ trials, let $$S_n$$ be the number of successes in the first $$n$$ trials, $$T_n$$ the number of successes in the last $$m$$ trials.

1. What is the distribution of $$S_n$$ ? Why ?
2. What is the distribution of $$T_m$$ ? Why ?
3. What is the distribution of $$S_n+T_m$$ ? Why ?
4. Are $$S_n$$ and $$T_m$$ independent ? Why ?
5. Are $$S_n$$ and $$T_{m+1}$$ independent ? Why ?
6. Are $$S_{n+1}$$ and $$T_{m}$$ independent ? Why ?

Based on [Pitman, p. 159, #10]

## Cards: Independence

A card is selected at random from a deck of 52 playing cards. If $$E$$ is the event that the card is a King and $$F$$ is the event that it is a heart. Show that $$E$$ and $$F$$ are independent events