A bag contains two coins, one fair and the other with two heads. You pick one coin at

Suppose you draw 2 cards from a standard deck of 52 cards. Find the probability that the second card is a spade given that the first card is a club.

Suppose you draw 2 cards from a standard deck of 52 cards. Find the probability that the second card is a spade given that the first card is a spade.

Suppose you draw 3 cards from a standard deck of 52 cards. Find the probability that the third card is a club given that the first two cards are spades.

Suppose you draw 3 cards from a standard deck of 52 cards. Find the probability that the third card is a club given that the first two cards are clubs.

An urn contains five blue and six green balls.You take two balls out of the urn, one after the other, without replacement. Find the probability that the second ball is green given that the first ball is blue.

An urn contains five green, six blue, and four red balls. You take three balls out of the urn, one after the other, without replacement. Find the probability that the third ball is green given that the first two balls were red.

A family has two children. The younger one is a girl. Find the probability that the other child is a girl as well.

A family has two children. One of their children is a girl. Find the probability that both children are girls.

You roll two fair dice. Find the probability that the first die is a 4 given that the sum is 7.

You roll two fair dice. Find the probability that the first die is a 5 given that the minimum of the two numbers is a 3.

You toss a fair coin three times. Find the probability that the first coin is heads given that at least one head occurred.

You toss a fair coin three times. Find the probability that at least two heads occurred given that the second toss resulted in heads.

You toss a fair coin four times. Find the probability that four heads occurred given that the first toss and the third toss resulted in heads.

You toss a fair coin four times. Find the proability of no more than three heads given that at least one toss resulted in heads.

A screening test for a disease shows a positive test result in 90% of all cases when the disease is actually present and in 15% of all cases when it is not. Assume that the prevalence of the disease is 1 in 100. If the test is administered to a randomly chosen individual, what is the probability that the result is negative?

A screening test for a disease shows a positive result in 92% of all cases when the disease is actually present and in 7% of all cases when it is not. Assume that the prevalence of the disease is 1 in 600. If the test is administered to a randomly chosen individual, what is the probability that the result is positive?

A patient underwent a diagnostic test for hypothyroidism. The diagnostic test correctly identifies patients who in fact have the disease in 93% of the cases and correctly identifies healthy patients in 81% of the cases. If 4 in 100 individuals have the disease, what is the probability that a test comes back negative?

A screening test for a disease shows a positive test result in 95% of all cases when the disease is actually present and in 20% of all cases when it is not. When the test was administered to a large number of people, 21.5% of the results were positive. What is the prevalence of the disease?

A drawer contains three bags numbered 13, respectively. Bag 1 contains three blue balls, bag 2 contains four green balls, and bag 3 contains two blue balls and one green ball. You choose one bag at random and take out one ball. Find the probability that the ball is blue.

A drawer contains six bags numbered 16, respectively. Bag i contains i blue balls and 2 green balls. You roll a fair die and then pick a ball out of the bag with the number shown on the die.What is the probability that the ball is blue?

You pick 2 cards from a standard deck of 52 cards. Find the probability that the second card is an ace. Compare this with the probability that the first card is an ace.

You pick 3 cards from a standard deck of 52 cards. Find the probability that the third card is an ace. Compare this with the probability that the first card is an ace.

Suppose that you have a batch of red-flowering pea plants of which 40% are of genotype CC and 60% of genotype Cc. You pick one plant at random and cross it with a white-flowering pea plant. Find the probability that the offspring of this crossing will have white flowers.

Suppose that you have a batch of red- and white-flowering pea plants, and suppose also that all three genotypes CC, Cc, and cc are equally represented in the batch. You pick one plant at random and cross it with a white-flowering pea plant.What is the probability that the offspring will have red flowers?

A bag contains two coins, one fair and the other with two heads.You pick one coin at random and flip it. Find the probability that the outcome is heads.

A drug company claims that a new headache drug will bring instant relief in 90% of all cases. If a person is treated with a placebo, there is a 20% chance that the person will feel instant relief. In a clinical trial, half the subjects are treated with the new drug and the other half receive the placebo. If an individual from this trial is chosen at random, what is the probability that the person will have experienced instant relief?

You are dealt 1 card from a standard deck of 52 cards. If A denotes the event that the card is a spade and if B denotes the event that the card is an ace, determine whether A and B are independent.

You are dealt 2 cards from a standard deck of 52 cards. If A denotes the event that the first card is an ace and B denotes the event that the second card is an ace, determine whether A and B are independent.

An urn contains five green and six blue balls. You take two balls out of the urn, one after the other, without replacement. If A denotes the event that the first ball is green and B denotes the event that the second ball is green, determine whether A and B are independent.

An urn contains four green and three blue balls. You take one ball out of the urn, note its color, and replace it. You then take a second ball out of the urn, note its color, and replace it. If A denotes the event that the first ball is green and B denotes the event that the second ball is green, determine whether A and B are independent.

Assume a 1:1 sex ratio. A family has three children. Find the probability of the event (a) A = {all children are girls} (b) B = {at least one boy} (c) C = {at least two girls} (d) D = {at most two boys}

Assume that 20% of a very common insect species in your study area is parasitized. Assume that insects are parasitized independently of each other. If you collect 10 specimens of this species, what is the probability that no more than 2 specimens in your sample are parasitized?

A multiple-choice question has four choices, and a test has a total of 10 multiple-choice questions. A student passes the test only if he or she answers all questions correctly. If the student guesses the answers to all questions randomly, find the probability that he or she will pass.

Assume that A and B are disjoint and that both events have positive probability. Are they independent?

Assume that the probability that an insect species lives more than five days is 0.1. Find the probability that, in a sample of size 10 of this species, at least one insect will still be alive after five days.

(a) Use a Venn diagram to show that (A B)c = Ac Bc (b) Use your result in (a) to show that if A and B are independent, then Ac and Bc are independent. (c) Use your result in (b) to show that if A and B are independent, then P(A B) = 1 P(Ac)P(Bc)

A screening test for a disease shows a positive result in 95% of all cases when the disease is actually present and in 10% of all cases when it is not. If the prevalence of the disease is 1 in 50 and an individual tests positive, what is the probability that the individual actually has the disease?

A screening test for a disease shows a positive result in 95% of all cases when the disease is actually present and in 10% of all cases when it is not. If a result is positive, the test is repeated. Assume that the second test is independent of the first test. If the prevalence of the disease is 1 in 50 and an individual tests positive twice, what is the probability that the individual actually has the disease?

A bag contains two coins, one fair and the other with two heads. You pick one coin at random and flip it. What is the probability that you picked the fair coin given that the outcome of the toss was heads?

You pick 2 cards from a standard deck of 52 cards. Find the probability that the first card was a spade given that the second card was a spade.

Suppose a woman has a hemophilic brother and one healthy son. Suppose furthermore that neither her mother nor her father were hemophilic but that her mother was a carrier for hemophilia. Find the probability that she is a carrier of the hemophilia gene.

(a) Given the pedigree, find the probability that the individual I-2 is a carrier of the hemophilia gene. (b) Given the pedigree, find the probability that II-3 is a carrier of the hemophilia gene.

(a) Given the pedigree, find the probability that II-3 is a carrier of the hemophilia gene. (b) Given the pedigree, find the probability that III-2 is a carrier of the hemophilia gene. (c) Given the pedigree, find the probability that II-2 is a carrier of the hemophilia gene.