I buy 67% of the apples for my pie-baking business from the cleverly named Apple Company #1 and the rest from the equally cleverly named Apple Company #2. I find that 28% of the apples from Apple Company #1 are unbruised, while 67% of the apples from Apple Company #2 are. Given that an apple is unbruised, what is the probability it came from Apple Company #1?

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At a certain company, the mentoring program and the community outreach program meet at the same time, so it is impossible for an employee to do both. If the probability that an employee participates in the mentoring program is 0.43, and the probability that an employee participates in the outreach program is 0.32, what is the probability that an employee does the mentoring program or the community outreach program?

A local TV station employs Desmorelda, “Mistress of the Zodiac”, as its weather forecaster. She predicts a rainy day on the basis of Lunar phases. From statistical data (which is available to Desmorelda too) it is known that the probability of a rainy day during the Full Moon is 0,478, during the Third Quarter – 0,513, during the New Moon – 0,499, and during the First Quarter – 0,506. All phases last equally and the whole Lunar cycle takes 28 days. What is the probability that on a certain day Desmorelda will predict rain?

A manufacturing company produces two types of smartphones, A and B. 30% of the smartphones produced are type A, while 70% are type B. The defect rate for type A smartphones is 2%, while the defect rate for type B smartphones is 4%. If a customer receives a defective smartphone, what is the probability that it is of type A? Show your calculations and reasoning.

In a certain town, 20% of the population owns a car, 30% owns a bicycle, and 50% owns neither. If a resident of the town is selected at random, what is the probability that they own both a car and a bicycle? Is it possible for the probability to be negative or greater than 1? How would the probabilities change if the town's population was doubled while keeping the same proportions of car and bicycle ownership?

Last year, at Northern Manufacturing Company, 200 people had colds during the year. One hundred fifty-five people who did no exercising had colds, and the remainder of the people with colds were involved in a weekly exercise program. Half of the 1,000 employees were involved in some type of exercise.   What is the probability that an employee will have a cold next year? Given that an employee is involved in an exercise program, what is the probability that he or she will get a cold next year? What is the probability that an employee who is not involved in an exercise program will get a cold next year? Are exercising and getting a cold independent events? Explain your answer."

Beck is allergic to peanuts. At a large dinner party one evening, he notices that the cheesecake options on the dessert table contain the flavors: 10 slices of chocolate,12 slices of caramel, 12 slices of peanut butter chocolate, and 8 slices of strawberry. The desserts are served to the guests at random.  what is the probability that becks cheesecake contains peanuts? what is the probability that becks dessert does not contain chocolate?