In this exercise we will use Bayes' Theorem to solve the

Suppose that E and F are events in a sample space and p(E) = 1/3, p(F) = 1/2, and p(E I F) = 2/5. Find p(F I E).

Suppose that E and F are events in a sample space and p(E) = 2/3, p(F) = 3/4,and p(F I E) = 5/8. Find peE I F).

Suppose that Frida selects a ball by first picking one of two boxes at random and then selecting a ball from this box at random. The first box contains two white balls and three blue balls, and the second box contains four white balls and one blue ball. What is the probability that Frida picked a ball from the first box if she has selected a blue ball?

Suppose that Ann selects a ball by first picking one of two boxes at random and then selecting a ball from this box. The first box contains three orange balls and four black balls, and the second box contains five orange balls and six black balls. What is the probability that Ann picked a ball from the second box if she has selected an orange ball?

Suppose that 8% of all bicycle racers use steroids, that a bicyclist who uses steroids tests positive for steroids 96% of the time, and that a bicyclist who does not use steroids tests positive for steroids 9% of the time. What is the probability that a randomly selected bicyclist who tests positive for steroids actually uses steroids?

When a test for steroids is given to soccer players, 98% of the players taking steroids test positive and 12% of the players not taking steroids test positive. Suppose that 5% of soccer players take steroids. What is the probability that a soccer player who tests positive takes steroids?

Suppose that a test for opium use has a 2% false positive rate and a 5% false negative rate. That is, 2% of people who do not use opium test positive for opium, and 5% of opium users test negative for opium. Furthermore, suppose that 1 % of people actually use opium. a) Find the probability that someone who tests negative for opium use does not use opium. b) Find the probability that someone who tests positive for opium use actually uses opium.

Suppose that one person in 10,000 people has a rare genetic disease. There is an excellent test for the disease; 99.9% of people with the disease test positive and only 0.02% who do not have the disease test positive. a) What is the probability that someone who tests positive has the genetic disease? b) What is the probability that someone who tests negative does not have the disease?

Suppose that 8% of the patients tested in a clinic are infected with HIV Furthermore, suppose that when a blood test for HIV is given, 98% of the patients infected with 6-32 HIV test positive and that 3% of the patients not infected with HIV test positive. What is the probability that a) a patient testing positive for mv with this test is infected with it? b) a patient testing positive for HIV with this test is not infected with it? c) a patient testing negative for mv with this test is infected with it? d) a patient testing negative for mv with this test is not infected with it?

Suppose that 4% of the patients tested in a clinic are infected with avian influenza. Furthermore, suppose that when a blood test for avian influenza is given, 97% of the patients infected with avian influenza test positive and that 2% of the patients not infected with avian influenza test positive. What is the probability that a) a patient testing positive for avian influenza with this test is infected with it? b) a patient testing positive for avian influenza with this test is not infected with it? c) a patient testing negative for avian influenza with this test is infected with it? d) a patient testing negative for avian influenza with this test is not infected with it?

An electronics company is planning to introduce a new camera phone. The company commissions a marketing report for each new product that predicts either the success or the failure of the product. Of new products introduced by the company, 60% have been successes. Furthermore, 70% of their successful products were predicted to be successes, while 40% of failed products were predicted to be successes. Find the probability this new camera phone will be successful if its success has been predicted

A space probe near Neptune communicates with Earth using bit strings. Suppose that in its transmissions it sends a l one-third of the time and a 0 two-thirds of the time. When a 0 is sent, the probability that it is received correctly is 0.9, and the probability that it is received incorrectly (as a I) is 0. 1 . When a 1 is sent, the probability that it is received correctly is 0.8, and the probability that it is received incorrectly (as a 0) is 0.2. a) Find the probability that a 0 is received. b) Use Bayes' Theorem to find the probability that a 0 was transmitted, given that a 0 was received

A space probe near Neptune communicates with Earth using bit strings. Suppose that in its transmissions it sends a l one-third of the time and a 0 two-thirds of the time. When a 0 is sent, the probability that it is received correctly is 0.9, and the probability that it is received incorrectly (as a I) is 0. 1 . When a 1 is sent, the probability that it is received correctly is 0.8, and the probability that it is received incorrectly (as a 0) is 0.2. a) Find the probability that a 0 is received. b) Use Bayes' Theorem to find the probability that a 0 was transmitted, given that a 0 was received

Suppose that E, FJ , F2 , and F3 are events from a sample space S and that Fl , F2, and F3 are mutually disjoint and their union is S. Find p(F2 I E) if peE I Fl ) = 2/7,p(E I F2) = 3/8, p(E I F3) = 1/2, p(FI) = 1/6, p(F2) = 1/2, and p(F3) = 1/3.

In this exercise we will use Bayes' Theorem to solve the Monty Hall puzzle (Example lO in Section 6. 1). Recall that in this puzzle you are asked to select one of three doors to open. There is a large prize behind one of the three doors and the other two doors are losers. After you select a door, Monty Hall opens one of the two doors you did not select that he knows is a losing door, selecting at random if both are losing doors. Monty asks you whether you would like to switch to this door. Suppose that the three doors in the puzzle are labeled 1, 2, and 3. Let W be the random variable whose value is the number of the winning door; assume that p(W = k) = 1/3 for k = 1 , 2, 3. Let M denote the random variable whose value is the number of the door that Monty opens. Suppose you choose door i. a) What is the probability that you will win the prize if the game ends before Monty asks you whether you want to change doors? b) Find p(M = j I W = k) for j = 1 , 2,3 and k = 1 , 2, 3. c) Use Bayes' Theorem to find p(W = j i M = k) where i and j and k are distinct values. d) Explain why the answer to part (c) tells you whether you should chang

Ramesh can get to work three different ways: by bicycle, by car, or by bus. Because of commuter traffic, there is a 50% chance that he will be late when he drives his car. When he takes the bus, which uses a special lane reserved for buses, there is a 20% chance that he will be late. The probability that he is late when he rides his bicycle is only 5%. Ramesh arrives late one day. His boss wants to estimate the probability that he drove his car to work that day. a) Suppose the boss assumes that there is a 1/3 chance that Ramesh takes each of the three ways he can get to work. What estimate for the probability that Ramesh drove his car does the boss obtain from Bayes' Theorem under this assumption? b) Suppose the boss knows that Ramesh drives 30% of the time, takes the bus only 10% of the time, and takes his bicycle 60% of the time. What estimate for the probability that Ramesh drove his car does the boss obtain from Bayes' Theorem using this information?

Prove Theorem 2, the extended form of Bayes' Theorem. That is, suppose that E is an event from a sample space S and that FI , F2, ... , Fn are mutually exclusive events such that U7=1 Fj = S. Assumethat p(E) =1= o and p(Fj) =1= 0 for i = 1 , 2, ... , n. Show that p(Fj I E) = n L... (E I F) (F)' /=I P / P / [Hint: Use the fact that E = U7=1 (E n Fj ).]

Suppose that a Bayesian spam filter is trained on a set of 500 spam messages and 200 messages that are not spam. The word "exciting" appears in 40 spam messages and in 25 messages that are not spam. Would an incoming message be rejected as spam if it contains the word "exciting" and the threshold for rejecting spam is 0.9?

Suppose that a Bayesian spam filter is trained on a set of 1000 spam messages and 400 messages that are not spam. The word "opportunity" appears in 175 spam messages and 20 messages that are not spam. Would an incoming message be rejected as spam if it contains the word "opportunity" and the threshold for rejecting a message is 0.9?

Would we reject a message as spam in Example 4 a) using just the fact that the word "undervalued" occurs in the message? b) using just the fact that the word "stock" occurs in the message?

Suppose that a Bayesian spam filter is trained on a set of 10,000 spam messages and 5000 messages that are not spam. The word "enhancement" appears in 1500 spam messages and 20 messages that are not spam, while the word "herbal" appears in 800 spam messages and 200 messages that are not spam. Estimate the probability that a received message containing both the words "enhancement" and "herbal" is spam. Will the message be rejected as spam if the threshold for rejecting spam is 0.9?

Suppose that we have prior information concerning whether a random incoming message is spam. In particular, suppose that over a time period, we find that s spam messages arrive and h messages arrive that are not spam. a) Use this information to estimate p(S), the probability that an incoming message is spam, and p(S), the probability an incoming message is not spam. b) Use Bayes' Theorem and part (a) to estimate the probability that an incoming message containing the word w is spam, where p(w) is the probability that w occurs in a spam message and q(w) is the probability that w occurs in a message that is not spam.

Suppose that EI and E2 are the events that an incoming mail message contains the words WI and W2, respectively. Assuming that E I and E 2 are independent events and that EI I S and E2 I S are independent events, where S is the event that an incoming message is spam, and that we have no prior knowledge regarding whether or not the message is spam, show that p(EI I S)P(E2 I S) p(EI I S)P(E2 I S) + p(EI I S)P(E2 I S)