Can the vectors in Problem be probability vectors? If not, why?

Problem 1. In a weekday, Jeremy will eat panda express 35% of the time, drink Starbucks 40% of the time. And if given he doesn't drink Starbucks, the probability he eats panda express is .5. Calculate the probability he will eats panda express and drink Starbucks at the same time.

Testing for a disease can be made more efficient by combining samples. If the samples from two people are combined and the mixture tests​ negative, then bothsamples are negative. On the other​ hand, one positive sample will always test​ positive, no matter how many negative samples it is mixed with. Assuming the probability of a single sample testing positive is 0.1​,find the probability of a positive result for two samples combined into one mixture. Is the probability low enough so that further testing of the individual samples is rarely​ necessary?       ​(Round to three decimal places as​ needed.) Is the probability low enough so that further testing of the individual samples is rarely​ necessary?     A. The probability is​ low, so further testing will be necessary for all of the combined mixtures.   B. The probability is​ low, so further testing of the individual samples will be a rarely necessary event.   C. The probability is not​ low, so further…

Problem 1.3 (Video 1.3, 1.4, Lecture Problem) Thor is trying to determine Loki's recent whereabouts. His approach is to go to a random planet and ask the local leader, "Has Loki visited this planet recently?" to which they will answer either "Yes" or "No." There is one problem: some of the planets are secretly controlled by Loki and the leaders will lie to protect him. The probability that a planet is controlled by Loki is 1/3. The probability that Loki has visited a planet recently is 1/2 if he controls it and 1/4 if he does not control it. If the planet is controlled by Loki, the leader will lie with probability 3/4. If the planet is not controlled by Loki, a leader will lie with probability 1/5 (for their own reasons that have nothing to do with Loki). (b) Determine the probability that Loki does not control the planet and Loki has visited recently and the leader answers "Yes," P[CcnVnY]. (c) Determine the probability that the leader answers "Yes" given that Loki has visited…