91Ó°ÊÓ

A student randomly guesses the answers to a four-question true-or-false \((T-F)\) quiz. Find the probability of each of the following events. (Hint: Do Exercise 12 first.) (a) \(E_{1}:\) "the student answers \(F\) on two of the four questions." (b) \(E_{2}\) : "the student answers \(F\) on at least two of the four questions." (c) \(E_{3}:\) "the student answers \(F\) on at most two of the four questions." (d) \(E_{4}\) : "the student answers \(F\) to the first two questions."

A student takes a 10 -question true-or-false quiz and randomly guesses the answer to each question. Suppose that a correct answer is worth 1 point and an incorrect answer is worth -0.5 points. Find the probability that the student (a) gets 10 points. (b) gets -5 points. (c) gets 8.5 points. (d) gets 8 or more points. (e) gets 5 points. (f) gets 7 or more points.

Suppose that the probability of giving birth to a boy and the probability of giving birth to a girl are both \(0.5 .\) Find the probability that in a family of four children, (a) all four children are girls. (b) there are two girls and two boys. (c) the youngest child is a girl. (d) the oldest child is a boy.

Ten professional basketball teams are participating in a draft lottery. (A draft lottery is a lottery to determine the order in which teams get to draft players.) Ten balls, each containing the name of one team (call them \(A, B, C, D, E\), \(F, G, H, I,\) and \(J\) for short), are placed in an urn and thoroughly mixed. Four balls are drawn, one at a time, from the urn. The four teams chosen get to draft first and in the order they are chosen. The remaining six teams have to draft in reverse order of season records. Find the probability that (a) \(A\) is the first team chosen. (b) \(A\) is one of the four teams chosen. (c) \(A\) is not one of the four teams chosen.

An honest coin is tossed 10 times in a row. The result of each toss ( \(H\) or \(T\) ) is observed. Find the probability of the event \(E="\) a \(T\) comes up at least once." (Hint: Find the probability of the complementary event.)

Imagine a game in which you roll an honest die three times. Find the probability of the event \(E="\) at least one of the rolls of the dice comes up a 6." (Hint: See Example 16.22.)

At Thomas Jefferson High School, the student body is divided by age as follows: \(7 \%\) of the students are \(14,22 \%\) of the students are \(15,24 \%\) of the students are \(16,23 \%\) of the students are \(17,19 \%\) of the students are 18 , and the rest of the students are \(19 .\) Find the average age of the students at Thomas Jefferson High School.

A basketball player shoots two consecutive free throws. Each free-throw is worth 1 point and has probability of success \(p=3 / 4\). Let \(X\) denote the number of points scored. Find the expected value of \(X\).

A box contains twenty \(\$ 1\) bills, ten \(\$ 5\) bills, five \(\$ 10\) bills, four \(\$ 20\) bills, and one \(\$ 100\) bill. You blindly reach into the box and draw a bill at random. What is the expected value of your draw?

Suppose that you roll a single die. If an odd number (1,3,0 5) comes up, you win the amount of your roll (\$1, \$3, or \$5 respectively). If an even number \((2,4,\) or 6\()\) comes up, you have to pay the house the amount of your roll \((\$ 2, \$ 4,\) or \(\$ 6\) respectively). (a) Find the expected payoff for this game. (b) Is this a fair game? Explain.