91Ó°ÊÓ

Probability Says... Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement of likelihood given. (The probability is usually a more exact measure of likelihood than is the verbal statement.) $$ \begin{array}{rrrrrrrr} 00.010 .450 .500 .550 .9910 & 0.01 & 0.45 & 0.50 & 0.55 & 0.99 & 1 \end{array} $$ (a) This event is impossible. It can never occur. (b) This event is certain. It will occur on every trial. (c) This event is very likely, but it will not occur once in a while in a long sequence of trials. (d) This event will occur slightly less often than not.

How Many Cups of Coffee? Choose an adult age 18 or over in the Unaited States at random and ask, "How many cups of coffee do you drink on average per day?" Call the response \(X\) for short. Based on a large sample survey, here is a probability model for the answer you will get: \({ }^{7}\) \begin{tabular}{l|ccccc} \hline Number & 0 & 1 & 2 & 3 & 4 or more \\ \hline Probability & \(0.36\) & \(0.26\) & \(0.19\) & \(0.08\) & \(0.11\) \\ \hline \end{tabular} (a) Verify that this is a valid finite probability model. (b) Describe the event \(X<4\) in words. What is \(P(X<4)\) ? (c) Express the event "have at least one cup of coffee on an average day" in terms of \(X\). What is the probability of this event?

Birth order. A couple plans to have three children. There are eight possible arrangements of girls and boys. For example, GGB means the first two children are girls and the third child is a boy. All eight arrangements are (approximately) equally likely. (a) Write down all eight arrangements of the sexes of three children. What is the probability of any one of these arrangements? (b) Let \(X\) be the number of girls the couple has. What is the probability that \(X=2 ?\) (c) Starting from your work in part (a), find the distribution of \(X\). That is, what values \(\operatorname{can} X\) take, and what are the probabilities for each value?

A taste test. A tea-drinking Canadian friend of yours claims to have a very refined palate. She tells you that she can tell if, in preparing a cup of tea, milk is first added to the cup and then hot tea poured into the cup, or the hot tea is first poured into the cup and then the milk is added. \({ }^{18}\) To test her claims, you prepare six cups of tea. Three have the milk added first and the other three the tea first. In a blind taste test, your friend tastes all six cups and is asked to identify the three that had the milk added first. (a) How many different ways are there to select three of the six cups? (Hint: See Example 12.8.) (b) If your friend is just guessing, what is the probability that she correctly identifies the three cups with the milk added first?