(A) Three marksmen fire simultaneously and independently at a target. What is the probability of the target being hit at least once, given that marksman one hits a target nine times out of ten, marksman two hits a target eight times out of ten while marksman three only hits a target one out of every two times. (B) Fifty teams compete in a student programming competition. It has been observed that 60% of the teams use the programming language C while the others use C++, and experience has shown that teams who program in C are twice as likely to win as those who use C++. Furthermore, ten teams who use C++ include a graduate student, while only four of those who use C include a graduate student. (a) What is the probability that the winning team programs in C? (b) What is the probability that the winning team programs in C and includes a graduate student? (c) What is the probability that the winning team includes a graduate student? (d) Given that the winning team includes a graduate student, what is the probability that team programmed in C? (C) A brand new light bulb is placed in a socket and the time it takes until it burns out is measured. Describe an appropriate sample space for this experiment. Use mathematical set notation to describe the following events: (a) A = thelight bulb lasts at least 100 hours. (b) B = thelight bulb lasts between 120 and 160 hours. (c) C = thelight bulb lasts less than 200 hours. (D) A university professor drives from his home in Cary to his university office in Raleigh each day. His car, which is rather old, fails to start one out of every eight times and he ends up taking his wife’s car. Furthermore, the rate of growth of Cary is so high that traffic problems are common. The professor finds that 70% of the time, traffic is so bad that he is forced to drive fast his preferred exit off the beltline, Western Boulevard, and take the next exit, Hillsborough street. What is the probability of seeing this professor driving to his office along Hillsborough street, in his wife’s car?

On the basis of a physical examination, a doctor determines the probability of no tumour   (event labelled C for ‘clear’), a benign tumour (B) or a malignant tumour (M) as 0.7, 0.2 and 0.1 respectively.

A further, in depth, test is conducted on the patient which can yield either a negative (N) result or positive (P). The test gives a negative result with probability 0.9 if no tumour is present (i.e. P(N|C) = 0.9). The test gives a negative result with probability 0.8 if there is a benign tumour and 0.2 if there is a malignant tumour.

(i) Given this information calculate the joint and marginal probabilities and display in the table below.

2. Obtain the posterior probability distribution for the patient when the test result is

            a) positive,   b) negative

4. Comment on how the test results change the doctor’s view of the presence of a tumour.

What is the probability that at least one of a pair of fair dice lands of 5, given that the sum of the dice is 9?

The probability that a house in an urban area will develop a leak is 6​%.If 44 houses are randomly​ selected, what is the probability that none of the houses will develop a​ leak? Round to the nearest thousandth.

Discuss briefly the purpose of probability distributions and expected values with examples

If the probability that a certain tennis player will serve an ace is 1/ 6 , what is the probability that he will serve exactly four aces out of six serves? (Assume that the six serves are independent. Round your answer to four decimal places.)

If the probability that a family will buy a vacation home in Manmi, Malibu, or Newport is 0.25, 0.10, 0.35, what is the probability the family will consummate one of these transactions? please show all wor with explanation.

What is the probability of finding the particle in the region a/4 < x < a/2? show your work.

A card is drawn at random from a deck of cards. What is the probability that

(a) it is a heart, given that it is red?

(b) it is higher than a 10, given that it is a heart? (Interpret J, Q, K, A as 11, 12, 13, 14.)

(c) it is a jack, given that it is red?

An experiment is picking a card from a fair deck. a.) What is the probability of picking a Jack given that the card is a face card? b.) What is the probability of picking a heart given that the card is a three? c.) What is the probability of picking a red card given that the card is an ace? d.) Are the events Jack and face card independent events? Why or why not? e.) Are the events red card and ace independent events? Why or why not?