Question: When discussing counting and probability, we often consider situations that may appear frivolous or of little practical value, such as tossing coins, choosing cards, or rolling dice. The reason is that these relatively simple examples serve as models for a wide variety of more complex situations in the real world. In light of this remark, comment on the relationship between your answer to exercise I and your answers to exercises II-III.
Exercise I: 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 HHT 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.
Exercise II: Suppose that each child born is equally likely to be a boy or a girl. Consider a family with exactly three children. Let BBG indicate that the first two children born are boys and the third child is a girl, let GBG 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 events in the next column 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.
Exercise III: Three people have been exposed to a certain illness. Once exposed, a person has a 50-50 chance of actually becoming ill.
a. What is the probability that exactly one of the people becomes ill?
b. What is the probability that at least two of the people become ill?
c. What is the probability that none of the three people becomes ill?