Let $A$ and $B$ be events with $\mathbb{P}(B)>0$. The conditional probability of $A$ given $B$ is defined to be $$\mathbb{P}(A\mid B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}$$

If $B$ is an event with $\mathbb{P}(B)>0$, then the conditional distribution of $X$ given $B$ is the probability measure on $(T,\mathcal{T})$ given by $$A\mapsto \mathbb{P}(X\in A\mid B),A\in \mathcal{T}$$

Suppose again that $B$ is an event with $\mathbb{P}(B)>0$. Then $A\mapsto \mathbb{P}(A\mid B)$ is a probability measure on $(S,\mathcal{S})$.

Suppose that $A$ and $B$ are events with $\mathbb{P}(B)>0$.

1. If $B\subseteq A$ then $\mathbb{P}(A\mid B)=1$.
2. If $A\subseteq B$ then $\mathbb{P}(A\mid B)=\mathbb{P}(A)/\mathbb{P}(B)$.
3. If $A$ and $B$ are disjoint then $\mathbb{P}(A\mid B)=0$.

Suppose that $A$, $B$, and $C$ are events with $\mathbb{P}(B\cap C)>0$. The probability of $A$ given $B$, relative to $\mathbb{P}(\cdot \mid C)$, is the same as the probability of $A$ given $B$ and $C$ (relative to $\mathbb{P}$). That is, $$\frac{\mathbb{P}(A\cap B\mid C)}{\mathbb{P}(B\mid C)}=\mathbb{P}(A\mid B\cap C)$$

Suppose that $A$ and $B$ are events with $\mathbb{P}(A)>0$ and $\mathbb{P}(B)>0$.

1. $\mathbb{P}(A\mid B)>\mathbb{P}(A)$ if and only if $\mathbb{P}(B\mid A)>\mathbb{P}(B)$ if and only if $\mathbb{P}(A\cap B)>\mathbb{P}(A)\mathbb{P}(B)$. In this case, $A$ and $B$ are positively correlated.
2. $\mathbb{P}(A\mid B)<\mathbb{P}(A)$ if and only if $\mathbb{P}(B\mid A)<\mathbb{P}(B)$ if and only if $\mathbb{P}(A\cap B)<\mathbb{P}(A)\mathbb{P}(B)$. In this case, $A$ and $B$ are negatively correlated.
3. $\mathbb{P}(A\mid B)=\mathbb{P}(A)$ if and only if $\mathbb{P}(B\mid A)=\mathbb{P}(B)$ if and only if $\mathbb{P}(A\cap B)=\mathbb{P}(A)\mathbb{P}(B)$. In this case, $A$ and $B$ are uncorrelated or independent.

Suppose that $A$ and $B$ are events in a random experiment.

1. $A$ and $B$ have the same correlation (positive, negative, or zero) as $A^c$ and $B^c$.
2. $A$ and $B$ have the opposite correlation as $A$ and $B^c$ (that is, positive-negative, negative-positive, or 0-0).

Suppose that $n\in \{2,3,\dots \}$ and that $(A_1,A_2,\dots ,A_n)$ is a sequence of events with $\mathbb{P}(A_1\cap A_2\cap \cdots \cap A_{n-1})>0$. Then $$\mathbb{P}(A_1\cap A_2\cap \cdots \cap A_n)=\mathbb{P}\left(A_1\right)\mathbb{P}(A_2\mid A_1)P(A_3\mid A_1\cap A_2)\cdots \mathbb{P}(A_n\mid A_1\cap A_2\cap \cdots \cap A_{n-1})$$

In the context of [8], if $E$ is another event, amd $\mathbb{P}(A_1\cap \cdots \cap A_{n-1}\cap E)>0$ then $$\mathbb{P}(A_1\cap A_2\cap \cdots \cap A_n\mid E)=\mathbb{P}(A_1\mid E)\mathbb{P}(A_2\mid A_1\cap E)P(A_3\mid A_1\cap A_2\cap E)\cdots \mathbb{P}(A_n\mid A_1\cap A_2\cap \cdots \cap A_{n-1}\cap E)$$

If $B$ is an event then $$\mathbb{P}(B)=\sum _{i\in I}\mathbb{P}(A_i)\mathbb{P}(B\mid A_i)$$

If $B$ is an event then $$\mathbb{P}(A_j\mid B)=\frac{\mathbb{P}(A_j)\mathbb{P}(B\mid A_j)}{\sum _{i\in I}\mathbb{P}(A_i)\mathbb{P}(B\mid A_i)},j\in I$$

If $E$ is another event with $\mathbb{P}(A_i\cap E)>0$ for $i\in I$ then $$\begin{array}{rl}\mathbb{P}(B\mid E)& =\sum _{i\in I}\mathbb{P}(A_i\mid E)\mathbb{P}(B\mid A_i\cap E)\\ \mathbb{P}(A_j\mid B\cap E)& =\frac{\mathbb{P}(A_j\mid E)\mathbb{P}(B\mid A_j\cap E)}{\sum _{i\in I}\mathbb{P}(A_i\cap E)\mathbb{P}(B\mid A_i\cap E)},j\in I\end{array}$$

Suppose that $A$ and $B$ are events in an experiment with $\mathbb{P}(A)=\frac{1}{3}$, $\mathbb{P}(B)=\frac{1}{4}$, $\mathbb{P}(A\cap B)=\frac{1}{10}$. Find each of the following:

1. $\mathbb{P}(A\mid B)$
2. $\mathbb{P}(B\mid A)$
3. $\mathbb{P}(A^c\mid B)$
4. $\mathbb{P}(B^c\mid A)$
5. $\mathbb{P}(A^c\mid B^c)$

Suppose that $A$, $B$, and $C$ are events in a random experiment with $\mathbb{P}(A\mid C)=\frac{1}{2}$, $\mathbb{P}(B\mid C)=\frac{1}{3}$, and $\mathbb{P}(A\cap B\mid C)=\frac{1}{4}$. Find each of the following:

1. $\mathbb{P}(B\setminus A\mid C)$
2. $\mathbb{P}(A\cup B\mid C)$
3. $\mathbb{P}(A^c\cap B^c\mid C)$
4. $\mathbb{P}(A^c\cup B^c\mid C)$
5. $\mathbb{P}(A^c\cup B\mid C)$
6. $\mathbb{P}(A\mid B\cap C)$

Suppose that $A$ and $B$ are events in a random experiment with $\mathbb{P}(A)=\frac{1}{2}$, $\mathbb{P}(B)=\frac{1}{3}$, and $\mathbb{P}(A\mid B)=\frac{3}{4}$.

1. Find $\mathbb{P}(A\cap B)$
2. Find $\mathbb{P}(A\cup B)$
3. Find $\mathbb{P}(B\cup A^c)$
4. Find $\mathbb{P}(B\mid A)$
5. Are $A$ and $B$ positively correlated, negatively correlated, or independent?

Open the conditional probability experiment.

1. Given $\mathbb{P}(A)$, $\mathbb{P}(B)$, and $\mathbb{P}(A\cap B)$, in the table, verify all of the other probabilities in the table.
2. Run the experiment 1000 times and compare the probabilities with the relative frequencies.

In a certain population, 30% of the persons smoke cigarettes and 8% have COPD (Chronic Obstructive Pulmonary Disease). Moreover, 12% of the persons who smoke have COPD.

1. What percentage of the population smoke and have COPD?
2. What percentage of the population with COPD also smoke?
3. Are smoking and COPD positively correlated, negatively correlated, or independent?

A company has 200 employees: 120 are women and 80 are men. Of the 120 female employees, 30 are classified as managers, while 20 of the 80 male employees are managers. Suppose that an employee is chosen at random.

1. Find the probability that the employee is female.
2. Find the probability that the employee is a manager.
3. Find the conditional probability that the employee is a manager given that the employee is female.
4. Find the conditional probability that the employee is female given that the employee is a manager.
5. Are the events female and manager positively correlated, negatively correlated, or indpendent?

Consider the experiment that consists of rolling 2 standard, fair dice and recording the sequence of scores $\mathit{X}=(X_1,X_2)$. Let $Y$ denote the sum of the scores. For each of the following pairs of events, find the probability of each event and the conditional probability of each event given the other. Determine whether the events are positively correlated, negatively correlated, or independent.

1. $\{X_1=3\}$, $\{Y=5\}$
2. $\{X_1=3\}$, $\{Y=7\}$
3. $\{X_1=2\}$, $\{Y=5\}$
4. $\{X_1=3\}$, $\{X_1=2\}$

In dice experiment, set $n=2$. Run the experiment 1000 times. Compute the empirical conditional probabilities corresponding to the conditional probabilities in exercise [19].

Consider again the experiment that consists of rolling 2 standard, fair dice and recording the sequence of scores $\mathit{X}=(X_1,X_2)$. Let $Y$ denote the sum of the scores, $U$ the minimum score, and $V$ the maximum score.

1. Find $\mathbb{P}(U=u\mid V=4)$ for the appropriate values of $u$.
2. Find $\mathbb{P}(Y=y\mid V=4)$ for the appropriate values of $y$.
3. Find $\mathbb{P}(V=v\mid Y=8)$ for appropriate values of $v$.
4. Find $\mathbb{P}(U=u\mid Y=8)$ for the appropriate values of $u$.
5. Find $\mathbb{P}[(X_1,X_2)=(x_1,x_2)\mid Y=8]$ for the appropriate values of $(x_1,x_2)$.

In the die-coin experiment, a standard, fair die is rolled and then a fair coin is tossed the number of times showing on the die. Let $N$ denote the die score and $H$ the event that all coin tosses result in heads.

1. Find $\mathbb{P}(H)$.
2. Find $\mathbb{P}(N=n\mid H)$ for $n\in \{1,2,3,4,5,6\}$.
3. Compare the results in (b) with $\mathbb{P}(N=n)$ for $n\in \{1,2,3,4,5,6\}$. In each case, note whether the events $H$ and $\{N=n\}$ are positively correlated, negatively correlated, or independent.

Run the die-coin experiment 1000 times. Let $H$ and $N$ be as defined in exercise [22].

1. Compute the empirical probability of $H$. Compare with the true probability in the previous exercise.
2. Compute the empirical probability of $\{N=n\}$ given $H$, for $n\in \{1,2,3,4,5,6\}$. Compare with the true probabilities in exercise [22].

Suppose that a bag contains 12 coins: 5 are fair, 4 are biased with probability of heads $\frac{1}{3}$; and 3 are two-headed. A coin is chosen at random from the bag and tossed.

1. Find the probability that the coin is heads.
2. Given that the coin is heads, find the conditional probability of each coin type.

In the coin-die experiment, a fair coin is tossed. If the coin lands tails, a fair die is rolled. If the coin lands heads, an ace-six flat die is tossed (faces 1 and 6 have probability $\frac{1}{4}$ each, while faces 2, 3, 4, and 5 have probability $\frac{1}{8}$ each). Let $H$ denote the event that the coin lands heads, and let $Y$ denote the score when the chosen die is tossed.

1. Find $\mathbb{P}(Y=y)$ for $y\in \{1,2,3,4,5,6\}$.
2. Find $\mathbb{P}(H\mid Y=y)$ for $y\in \{1,2,3,4,5,6,\}$.
3. Compare each probability in part (b) with $\mathbb{P}(H)$. In each case, note whether the events $H$ and $\{Y=y\}$ are positively correlated, negatively correlated, or independent.

Run the coin-die experiment 1000 times. Let $H$ and $Y$ be as defined in [25].

1. Compute the empirical probability of $\{Y=y\}$, for each $y$, and compare with the true probability in exercise [25]
2. Compute the empirical probability of $H$ given $\{Y=y\}$ for each $y$, and compare with the true probability in exercise [25].

Consider the card experiment that consists of dealing 2 cards from a standard deck and recording the sequence of cards dealt. For $i\in \{1,2\}$, let $Q_i$ be the event that card $i$ is a queen and $H_i$ the event that card $i$ is a heart. For each of the following pairs of events, compute the probability of each event, and the conditional probability of each event given the other. Determine whether the events are positively correlated, negatively correlated, or independent.

1. $Q_1$, $H_1$
2. $Q_1$, $Q_2$
3. $Q_2$, $H_2$
4. $Q_1$, $H_2$

In the card experiment, set $n=2$. Run the experiment 500 times. Compute the conditional relative frequencies corresponding to the conditional probabilities in exercise [27].

Consider the card experiment that consists of dealing 3 cards from a standard deck and recording the sequence of cards dealt. Find the probability of the following events:

1. All three cards are all hearts.
2. The first two cards are hearts and the third is a spade.
3. The first and third cards are hearts and the second is a spade.

In the card experiment, set $n=3$ and run the simulation 1000 times. Compute the empirical probability of each event in exercise [29] and compare with the true probability.

In Buffon's coin experiment,

1. Find $\mathbb{P}(Y>0\mid X<Y)$
2. Find the conditional distribution of $(X,Y)$ given that the coin does not touch the sides of the square.

Run Buffon's coin experiment 500 times. Compute the empirical probability that $Y>0$ given that $X<Y$ and compare with the probability in exercise [31].

In the conditional probability experiment, the random points are uniformly distributed on the rectangle $S$. Move and resize events $A$ and $B$ and note how the probabilities change. For each of the following configurations, run the experiment 1000 times and compare the relative frequencies with the true probabilities.

1. $A$ and $B$ in general position
2. $A$ and $B$ disjoint
3. $A\subseteq B$
4. $B\subseteq A$

A plant has 3 assembly lines that produces memory chips. Line 1 produces 50% of the chips and has a defective rate of 4%; line 2 has produces 30% of the chips and has a defective rate of 5%; line 3 produces 20% of the chips and has a defective rate of 1%. A chip is chosen at random from the plant.

1. Find the probability that the chip is defective.
2. Given that the chip is defective, find the conditional probability for each line.

Suppose that a bit (0 or 1) is sent through a noisy communications channel. Because of the noise, the bit sent may be received incorrectly as the complementary bit. Specifically, suppose that if 0 is sent, then the probability that 0 is received is 0.9 and the probability that 1 is received is 0.1. If 1 is sent, then the probability that 1 is received is 0.8 and the probability that 0 is received is 0.2. Finally, suppose that 1 is sent with probability 0.6 and 0 is sent with probability 0.4. Find the probability that

1. 1 was sent given that 1 was received
2. 0 was sent given that 0 was received

Suppose that $T$ denotes the lifetime of a light bulb (in 1000 hour units), and that $T$ has the following exponential distribution, defined for measurable $A\subseteq [0,\mathrm{\infty })$: $$\mathbb{P}(T\in A)={\int }_A{e}^{-t}dt$$

1. Find $\mathbb{P}(T>3)$
2. Find $\mathbb{P}(T>5\mid T>2)$

Suppose again that $T$ denotes the lifetime of a light bulb (in 1000 hour units), but that $T$ is uniformly distributed on the interal $[0,10]$.

1. Find $\mathbb{P}(T>3)$
2. Find $\mathbb{P}(T>5\mid T>2)$

Suppose that a person is chosen at random from the population. Let $A$, $B$, $AB$, and $O$ be the events that the person is type $A$, type $B$, type $AB$, and type $O$ respectively. Let $H$ be the event that the person is homozygous, and let $D$ denote the event that the person has an $o$ allele. Find each of the following:

1. $\mathbb{P}(A)$, $\mathbb{P}(B)$, $\mathbb{P}(AB)$, $\mathbb{P}(O)$, $\mathbb{P}(H)$, $\mathbb{P}(D)$
2. $P(A\cap H)$, $P(A\mid H)$, $P(H\mid A)$. Are the events $A$ and $H$ positively correlated, negatively correlated, or independent?
3. $P(B\cap H)$, $P(B\mid H)$, $P(H\mid B)$. Are the events $B$ and $H$ positively correlated, negatively correlated, or independent?
4. $P(A\cap D)$, $P(A\mid D)$, $P(D\mid A)$. Are the events $A$ and $D$ positively correlated, negatively correlated, or independent?
5. $P(B\cap D)$, $P(B\mid D)$, $P(D\mid B)$. Are the events $B$ and $D$ positively correlated, negatively correlated, or independent?
6. $P(H\cap D)$, $P(H\mid D)$, $P(D\mid H)$. Are the events $H$ and $D$ positively correlated, negatively correlated, or independent?

Suppose that a green-pod plant and a yellow-pod plant are bred together. Suppose further that the green-pod plant has a $\frac{1}{4}$ chance of carrying the recessive yellow-pod allele.

1. Find the probability that a child plant will have green pods.
2. Given that a child plant has green pods, find the updated probability that the green-pod parent has the recessive allele.

Suppose that two green-pod plants are bred together. Suppose further that with probability $\frac{1}{3}$ neither plant has the recessive allele, with probability $\frac{1}{2}$ one plant has the recessive allele, and with probability $\frac{1}{6}$ both plants have the recessive allele.

1. Find the probability that a child plant has green pods.
2. Given that a child plant has green pods, find the updated probability that both parents have the recessive gene.

Suppose that in a certain population, 50% are male and 50% are female. Moreover, suppose that 10% of males are color blind but only 1% of females are color blind.

1. Find the percentage of color blind persons in the population.
2. Find the percentage of color blind persons that are male.

A man and a woman do not have a certain sex-linked hereditary disorder, but the woman has a $\frac{1}{3}$ chance of being a carrier.

1. Find the probability that a son born to the couple will be normal.
2. Find the probability that a daughter born to the couple will be a carrier.
3. Given that a son born to the couple is normal, find the updated probability that the mother is a carrier.

Urn 1 contains 4 red and 6 green balls while urn 2 contains 7 red and 3 green balls. An urn is chosen at random and then a ball is chosen at random from the selected urn.

1. Find the probability that the ball is green.
2. Given that the ball is green, find the conditional probability that urn 1 was selected.

Urn 1 contains 4 red and 6 green balls while urn 2 contains 6 red and 3 green balls. A ball is selected at random from urn 1 and transferred to urn 2. Then a ball is selected at random from urn 2.

1. Find the probability that the ball from urn 2 is green.
2. Given that the ball from urn 2 is green, find the conditional probability that the ball from urn 1 was green.

An urn initially contains 6 red and 4 green balls. A ball is chosen at random from the urn and its color is recorded. It is then replaced in the urn and 2 new balls of the same color are added to the urn. The process is repeated. Find the probability of each of the following events:

1. Balls 1 and 2 are red and ball 3 is green.
2. Balls 1 and 3 are red and ball 2 is green.
3. Ball 1 is green and balls 2 and 3 are red.
4. Ball 2 is red.
5. Ball 1 is red given that ball 2 is red.

An urn initially contains 6 red and 4 green balls. A ball is chosen at random from the urn and its color is recorded. It is then replaced in the urn and two new balls of the other color are added to the urn. The process is repeated. Find the probability of each of the following events:

1. Balls 1 and 2 are red and ball 3 is green.
2. Balls 1 and 3 are red and ball 2 is green.
3. Ball 1 is green and balls 2 and 3 are red.
4. Ball 2 is red.
5. Ball 1 is red given that ball 2 is red.

Typical examples of events of interest and corresponding diagnostic tests:

1. The event is that a person has a certain disease and the test is a blood test for the disease.
2. The event is that a woman is pregnant and the test is a home pregnancy test.
3. The event is that a person is lying and the test is a lie-detector test.
4. The event is that a device is defective and the test consists of a sensor reading.
5. The event is that a missile is in a certain region of airspace and the test consists of radar signals.
6. The event is that a person has committed a crime, and the test is a jury trial with evidence presented for and against the event.

Let $T$ be the event that the test is positive for the occurrence of $A$.

1. The conditional probability $\mathbb{P}(T\mid A)$ is the sensitivity of the test. The complementary probability $\mathbb{P}(T^c\mid A)=1-\mathbb{P}(T\mid A)$ is the false negative probability.
2. The conditional probability $\mathbb{P}(T^c\mid A^c)$ is the specificity of the test. The complementary probability $\mathbb{P}(T\mid A^c)=1-\mathbb{P}(T^c\mid A^c)$ is the false positive probability.

The probability that the event occurs, given a positive test is $$\mathbb{P}(A\mid T)=\frac{\mathbb{P}(A)\mathbb{P}(T\mid A)}{\mathbb{P}(A)\mathbb{P}(T\mid A)+\mathbb{P}(A^c)\mathbb{P}(T\mid A^c)}$$ The probability that the event does not occur, given a negative test is $$\mathbb{P}(A^c\mid T^c)=\frac{\mathbb{P}(A^c)\mathbb{P}(T^c\mid A^c)}{\mathbb{P}(A)\mathbb{P}(T^c\mid A)+\mathbb{P}(A^c)\mathbb{P}(T^c\mid A^c)}$$

$P$ as a function of $p$ is given by $$P=\frac{ap}{(a+b-1)p+(1-b)},p\in [0,1]$$

1. $P$ increases continuously from 0 to 1 as $p$ increases from 0 to 1.
2. $P$ is concave downward if $a+b>1$. In this case $A$ and $T$ are positively correlated.
3. $P$ is concave upward if $a+b<1$. In this case $A$ and $T$ are negatively correlated.
4. $P=p$ if $a+b=1$. In this case, $A$ and $T$ are uncorrelated (independent).

Suppose that a diagnostic test has sensitivity 0.99 and specificity 0.95. Find $\mathbb{P}(A\mid T)$ for each of the following values of $\mathbb{P}(A)$:

1. 0.001
2. 0.01
3. 0.2
4. 0.5
5. 0.7
6. 0.9

A woman initially believes that there is an even chance that she is or is not pregnant. She takes a home pregnancy test with sensitivity 0.95 and specificity 0.90 (which are reasonable values for a home pregnancy test). Find the updated probability that the woman is pregnant in each of the following cases.

1. The test is positive.
2. The test is negative.

Suppose that 70% of defendants brought to trial for a certain type of crime are guilty. Moreover, historical data show that juries convict guilty persons 80% of the time and convict innocent persons 10% of the time. Suppose that a person is tried for a crime of this type. Find the updated probability that the person is guilty in each of the following cases:

1. The person is convicted.
2. The person is acquitted.

The Check Engine light on your car has turned on. Without the information from the light, you believe that there is a 10% chance that your car has a serious engine problem. You learn that if the car has such a problem, the light will come on with probability 0.99, but if the car does not have a serious problem, the light will still come on, under circumstances similar to yours, with probability 0.3. Find the updated probability that you have an engine problem.

The standard test for HIV is the ELISA (Enzyme-Linked Immunosorbent Assay) test. It has sensitivity and specificity of 0.999. Suppose that a person is selected at random from a population in which 1% are infected with HIV, and given the ELISA test. Find the probability that the person has HIV in each of the following cases:

1. The test is positive.
2. The test is negative.

The PSA test for prostate cancer is based on a blood marker known as the Prostate Specific Antigen. An elevated level of PSA is evidence for prostate cancer. To have a diagnostic test, in the sense that we are discussing here, we must decide on a definite level of PSA, above which we declare the test to be positive. A positive test would typically lead to other more invasive tests (such as biopsy) which, of course, carry risks and cost. The PSA test with cutoff 2.6 ng/ml has sensitivity 0.40 and specificity 0.81. The overall incidence of prostate cancer among males is 156 per 100000. Suppose that a man, with no particular risk factors, has the PSA test. Find the probability that the man has prostate cancer in each of the following cases:

1. The test is positive.
2. The test is negative.

The rapid molecular test for COVID-19 has sensitivity 0.952 and specificity of 0.0.980. Suppose that a person is selected at random from a population in which 10% are infected with COVID-19, and given the rapid test. Find the probability that the person is infected with COVID-19 in each of the following cases:

1. The test is positive.
2. The test is negative.

For the M&M data set, find the empirical probability that a bag has at least 10 reds, given that the weight of the bag is at least 48 grams.

Consider the Cicada data.

1. Find the empirical probability that a cicada weighs at least 0.25 grams given that the cicada is male.
2. Find the empirical probability that a cicada weighs at least 0.25 grams given that the cicada is the tredecula species.