91Ó°ÊÓ

Suppose two dice (one red, one green) are rolled. Consider the following events. A: the red die shows \(1 ; B:\) the numbers add to \(4 ; C:\) at least one of the numbers is \(1 ;\) and \(D:\) the numbers do not add to 11. Express the given event in symbols and say how many elements it contains. HINT [See Example 5.] The red die shows 1 and the numbers add to \(4 .\)

A tetrahedral die has four faces, numbered \(1-4\). If the die is weighted in such a way that each number is twice as likely to land facing down as the next number (1 twice as likely as 2 , 2 twice as likely as 3 , and so on) what is the probability distribution for the face landing down?

A standard piano keyboard has 88 different keys. Find the probability that a cat, jumping on 4 keys in sequence and at random (possibly with repetition), will strike the first four notes of Beethoven's Fifth Symphony. (Leave your answer as a formula.)

Require the use of counting techniques from the last chapter. A bag contains three red marbles, two green ones, one fluorescent pink one, two yellow ones, and two orange ones. Suzan grabs four at random. Find the probabilities of the indicated events. She gets at least two red ones, given that she gets at least one green one.

Say whether the given pairs of events are independent, mutually exclusive, or neither: \(A\) : Your new skateboard design is a success. \(B\) : Your new skateboard design is a failure.

The Sorry State Lottery requires you to select five different numbers from 0 through 49 . (Order is not important.) You are a Big Winner if the five numbers you select agree with those in the drawing, and you are a Small-Fry Winner if four of your five numbers agree with those in the drawing. What is the probability of being a Big Winner? What is the probability of being a Small-Fry Winner? What is the probability that you are either a Big Winner or a Small-Fry winner?

The Sad State Lottery requires you to select a sequence of three different numbers from 0 through 49 . (Order is important.) You are a winner if your sequence agrees with that in the drawing, and you are a booby prize winner if your selection of numbers is correct, but in the wrong order. What is the probability of being a winner? What is the probability of being a booby prize winner? What is the probability that you are either a winner or a booby prize winner?

Two dice (one red and one green) are rolled, and the numbers that face up are observed. Test the given pairs of events for independence. A \text { : Exactly one die is } 1 ; B \text { : The sum is even. }

Use the given information to find the indicated probability. \(A\) and \(B\) are mutually exclusive. \(P(A)=.4, P(B)=.4\). Find \(P\left((A \cup B)^{\prime}\right)\).

based on the following table, which shows the results of a survey of authors by a (fictitious) publishing company. HINT [See Example 5.] $$ \begin{array}{|r|c|c|c|} \hline & \text { New Authors } & \text { Established Authors } & \text { Total } \\ \hline \text { Successful } & 5 & 25 & 30 \\ \hline \text { Unsuccessful } & 15 & 55 & 70 \\ \hline \text { Total } & 20 & 80 & 100 \\ \hline \end{array} $$ Consider the following events: \(S:\) an author is successful; \(U\) : an author is unsuccessful; \(N:\) an author is new; and \(E:\) an author is established. Describe the events \(S \cap N\) and \(S \cup N\) in words. Use the table to compute \(n(S \cap N)\) and \(n(S \cup N)\).