In a high school graduating class of 100 students, 54 studied mathematics, 69 studied history, and 35 studied both mathematics and history. If one of these students is selected at random, find the probability that (a) the student took mathematics or history; (b) the student did not take either of these subjects; (c) the student took history but not mathematics.

In a poker hand consisting of 5 cards, find the probability of holding (a) 3 aces; (b) 4 hearts and 1 club.

Interest centers around the life of an electronic component. Suppose it is known that the probability that the component survives for more than 6000 hours is 0.42. Suppose also that the probability that the component survives no longer than 4000 hours is 0.04. (a) What is the probability that the life of the component is less than or equal to 6000 hours? (b) What is the probability that the life is greater than 4000 hours?

Suppose that in a senior college class of 500 students it is found that 210 smoke, 258 drink alcoholic beverages, 216 eat between meals, 122 smoke and drink alcoholic beverages, 83 eat between meals and drink alcoholic beverages, 97 smoke and eat between meals, and 52 engage in all three of these bad health practices. If a member of this senior class is selected at random, find the probability that the student (a) smokes but does not drink alcoholic beverages; (b) eats between meals and drinks alcoholic beverages but does not smoke; (c) neither smokes nor eats between meals.

A pair of fair dice is tossed. Find the probability of getting (a) a total of 8; (b) at most a total of 5.

If a letter is chosen at random from the English alphabet, find the probability that the letter (a) is a vowel exclusive of $y$; (b) is listed somewhere ahead of the letter $j$; (c) is listed somewhere after the letter $g$.

It is common in many industrial areas to use a filling machine to fill boxes full of product. This occurs in the food industry as well as other areas in which the product is used in the home, for example, detergent. These machines are not perfect, and indeed they may A, fill to specification, B, underfill, and C, overfill. Generally, the practice of underfilling is that which one hopes to avoid. Let P(B) = 0.001 while P(A) = 0.990. (a) Give P(C). (b) What is the probability that the machine does not underfill? (c) What is the probability that the machine either overfills or underfills?

If each coded item in a catalog begins with 3 distinct letters followed by 4 distinct nonzero digits, find the probability of randomly selecting one of these coded items with the first letter a vowel and the last digit even.

From past experience, a stockbroker believes that under present economic conditions a customer will invest in tax-free bonds with a probability of 0.6, will invest in mutual funds with a probability of 0.3, and will invest in both tax-free bonds and mutual funds with a probability of 0.15. At this time, find the probability that a customer will invest (a) in either tax-free bonds or mutual funds; (b) in neither tax-free bonds nor mutual funds.

A California study concluded that following 7 simple health rules can extend a man’s life by 11 years on the average and a woman’s life by 7 years. These 7 rules are as follows: no smoking, get regular exercise, use alcohol only in moderation, get 7 to 8 hours of sleep, maintain proper weight, eat breakfast, and do not eat between meals. In how many ways can a person adopt 5 of these rules to follow (a) if the person presently violates all 7 rules? (b) if the person never drinks and always eats breakfast?

A class in advanced physics is composed of 10 juniors, 30 seniors, and 10 graduate students. The final grades show that 3 of the juniors, 10 of the seniors, and 5 of the graduate students received an A for the course. If a student is chosen at random from this class and is found to have earned an A, what is the probability that he or she is a senior?

A box contains 500 envelopes, of which 75 contain $100 in cash, 150 contain$25, and 275 contain $10. An envelope may be purchased for$25. What is the sample space for the different amounts of money? Assign probabilities to the sample points and then find the probability that the first envelope purchased contains less than $100.

An automobile manufacturer is concerned about a possible recall of its best-selling four-door sedan. If there were a recall, there is a probability of 0.25 of a defect in the brake system, 0.18 of a defect in the transmission, 0.17 of a defect in the fuel system, and 0.40 of a defect in some other area. (a) What is the probability that the defect is the brakes or the fueling system if the probability of defects in both systems simultaneously is 0.15? (b) What is the probability that there are no defects in either the brakes or the fueling system?

Dom’s Pizza Company uses taste testing and statistical analysis of the data prior to marketing any new product. Consider a study involving three types of crusts (thin, thin with garlic and oregano, and thin with bits of cheese). Dom’s is also studying three sauces (standard, a new sauce with more garlic, and a new sauce with fresh basil). (a) How many combinations of crust and sauce are involved? (b) What is the probability that a judge will get a plain thin crust with a standard sauce for his first taste test?