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In Exercises \(7-10\) , determine whether the events are independent. (See Examples 1 and \(2 . )\) A student is taking a multiple-choice test where each question has four choices. The student randomly guesses the answers to the five-question test. Use a sample space to determine whether guessing Question 1 correctly and Question 2 correctly are independent events.

In Exercises \(7-10\) , determine whether the events are independent. (See Examples 1 and \(2 . )\) A vase contains four white roses and one red rose. You randomly select two roses to take home. Use a sample space to determine whether randomly selecting a white rose first and randomly selecting a white rose second are independent events.

Of 162 students honored at an academic awards banquet, 48 won awards for mathematics and 78 won awards for English. There are 14 students who won awards for both mathematics and English. A newspaper chooses a student at random for an interview. What is the probability that the student interviewed won an award for English or mathematics?

In Exercises \(9-12\) , calculate the probability of flipping a coin 20 times and getting the given number of heads. 18

PROBLEM SOLVING You play a game that involves drawing two numbers from a hat. There are 25 pieces of paper numbered from 1 to 25 in the hat. Each number is replaced after it is drawn. Find the probability that you will draw the 3 on your first draw and a number greater than 10 on your second draw.

In Exercises 13 and \(14,\) you roll a six-sided die. Find \(P(A \) or \(B) .\) Event \(A :\) Roll a \(6 .\) Event \(B :\) Roll a prime number.

In Exercises 13 and \(14,\) you roll a six-sided die. Find \(P(A \) or \(B) .\) Event \(A :\) Roll an odd number. Event \(B :\) Roll a number less than 5 .

MODELING WITH MATHEMATICS According to a survey, 48\(\%\) of adults in the United States believe that Unidentified Flying Objects (UFOs) are observing our planet. You ask 8 randomly chosen adults whether they believe UFOs are watching Earth. a. Draw a histogram of the binomial distribution for your survey. b. What is the most likely outcome of your survey? c. What is the probability that at most 3 people believe UFOs are watching Earth?

ERROR ANALYSIS Events \(A\) and \(B\) are independent. Describe and correct the error in finding \(P(A \text { and } B)\) . $$\begin{array}{l}{P(A)=0.6 \quad P(B)=0.2} \\ {P(A \text { and } B)=0.6+0.2=0.8}\end{array}$$

ERROR ANALYSIS A shelf contains 3 fashion magazines and 4 health magazines. You randomly choose one to read, set it aside, and randomly choose another for your friend to read. Describe and correct the error in finding the probability that both events \(A\) and \(B\) occur. Event \(A :\) The first magazine is fashion. Event \(B\) : The second magazine is health. $$\begin{array}{ll}{P(A)=\frac{3}{7}}{P(B | A)=\frac{4}{7}} \\ {P(A \text { and } B)=\frac{3}{7} \cdot \frac{4}{7}=\frac{12}{49}} & {=0.245}\end{array}$$