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Two new drugs are to be tested using a group of 60 laboratory mice, each tagged with a number for identification purposes. Drug \(A\) is to be given to 22 mice, drug \(B\) is to be given to another 22 mice, and the remaining 16 mice are to be used as controls. How many ways can the assignment of treatments to mice be made? (A single assignment involves specifying the treatment for each mouse-whether drug \(A\), drug \(B\), or no drug.)

Suppose there are three routes from North Point to Boulder Creek, two routes from Boulder Creek to Beaver Dam, two routes from Beaver Dam to Star Lake, and four routes directly from Boulder Creek to Star Lake. (Draw a sketch.) a. How many routes from North Point to Star Lake pass through Beaver Dam? b. How many routes from North Point to Star Lake bypass Beaver Dam?

A group of eight people are attending the movies together. a. Two of the eight insist on sitting side-by-side. In how many ways can the eight be seated together in a row? b. Two of the people do not like each other and do not want to sit side-by- side. Now how many ways can the eight be seated together in a row?

Suppose that a coin is tossed three times and the side showing face up on each toss is noted. Suppose also that on each toss heads and tails are equally likely. Let \(H H T\) indicate the outcome heads on the first two tosses and tails on the third, THT the outcome tails on the first and third tosses and heads on the second, and so forth. a. List the eight elements in the sample space whose outcomes are all the possible head-tail sequences obtained in the three tosses. b. Write each of the following events as a set and find its probability: (i) The event that exactly one toss results in a head. (ii) The event that at least two tosses result in a head. (iii) The event that no head is obtained.

a. A bit string is a finite sequence of 0 's and 1 's. How many bit strings have length 8 ? b. How many bit strings of length 8 begin with three 0 's? c. How many bit strings of length 8 begin and end with a 1 ? d. In Section \(1.5\) we showed how integers can be represented by strings of 0 's and 1 's inside a digital computer. In fact, through various coding schemes, strings of 0 's and l's can be used to represent all kinds of symbols. One commonly used code is the Extended Binary-Coded Decimal Interchange Code (EBCDIC) in which each symbol has an 8-bit representation. How many distinct symbols can be represented by this code?

How many pairs of two distinct integers chosen from the set \(\\{1,2,3, \ldots, 101\\}\) have a sum that is even?

Suppose that each child born is equally likely to be a boy or a girl. Consider a family with exactly three children. Let \(B B G\) indicate that the first two children born are boys and the third child is a girl, let \(G B G\) indicate that the first and third children born are girls and the second is a boy, and so forth. a. List the eight elements in the sample space whose outcomes are all possible genders of the three children. b. Write each of the following events as a set and find its probability. (i) The event that exactly one child is a girl. (ii) The event that at least two children are girls. (iii) The event that no child is a girl.

A coin is tossed ten times. In each case the outcome \(H\) (for heads) or \(T\) (for tails) is recorded. (One possible outcome of the ten tossings is denoted \(T H H T T T H T T H .)\) a. What is the total number of possible outcomes of the coin-tossing experiment? b. In how many of the possible outcomes are exactly five heads obtained? c. In how many of the possible outcomes are at least eight heads obtained? d. In how many of the possible outcomes is at least one head obtained? e. In how many of the possible outcomes is at most one head obtained?

A coin is tossed four times. Each time the result \(H\) for heads or \(T\) for tails is recorded. An outcome of HHTT means that heads were obtained on the first two tosses and tails on the second two. Assume that heads and tails are equally likely on each toss. a. How many distinct outcomes are possible? b. What is the probability that exactly two heads occur? c. What is the probability that exactly one head occurs?

Use the axioms for probability and mathematical induction to prove that for all integers \(n \geq 2\), if \(A_{1}, A_{2}, A_{3}, \ldots, A_{n}\) are any mutually disjoint events in a sample space \(S\), then $$ P\left(A_{1} \cup A_{2} \cup A_{3} \cup \cdots \cup A_{n}\right)=\sum_{k=1}^{n} P\left(A_{k}\right) $$