Exam 11: General Rules of Probability

In a particular game, a six-sided fair die is tossed. If the number of spots showing is six, you win $6; if the number of spots showing is five, you win $3; if the number of spots showing is four, you win $2; and if the number of spots showing is three, you win $1. If the number of spots showing is one or two, you win nothing. You are going to play the game twice. The probability that you win at least $9 in total on the two plays of the game is:

A group of freshmen at a local university consider joining the equestrian team. Thirty-five percent of students choose Western riding, 45% choose dressage, and 50% choose jumping. Twenty percent choose both dressage and jumping, while 10% choose Western and dressage. No one chooses Western and jumping. There are no horses suitable for two styles, and each student is assigned to one horse. What is the probability that a student chooses dressage or jumping?

A group of freshmen at a local university consider joining the equestrian team. Thirty-five percent of students choose Western riding, 45% choose dressage, and 40% choose jumping. Fifteen percent choose both dressage and jumping, while 15% choose Western and dressage. No one chooses Western and jumping. There are no horses suitable for two styles, and each student is assigned to one horse. If two students decide to join the team, what is the probability that one student joins jumping only and the other student joins the Western team, if they independently decide?

A system has two components that operate in parallel, as shown in the following diagram. Because the components operate in parallel, at least one of the components must function properly if the system is to function properly. The probability of failure for components 1 and 2, during one period of operation, is 0.2 and 0.03, respectively. Let F1 denote the event that component 1 fails during one period of operation and F2 denote the event that component 2 fails during one period of operation. The component failures are independent. The probability that the system functions properly during one period of operation is closest to:

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which requires meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The study also shows that the probability a student gets no degree is P(no) = 0.2. Some probability calculations show the probability of getting a Bachelor of Arts and Science degree to be P(Arts & Science) = 0.1. The probability of getting a Bachelor of Arts and Science degree, given that a student is getting a Bachelor of Science degree, is:

A group of freshmen at a local university consider joining the equestrian team. Thirty-five percent of students choose Western riding, 45% choose dressage, and 40% choose jumping. Fifteen percent choose both dressage and jumping, while 10% choose Western and dressage. No one chooses Western and jumping. There are no horses suitable for two styles, and each student is assigned to one horse. If two students decide to join the team, what is the probability that one student joins the dressage team and the other student does not?

A group of college DJs surveyed students to find out what music to plan for their upcoming parties. Thirty percent of the students preferred dubstep, 25% of the students liked trance music, and 20% wanted to hear only house music. Fifteen percent of the respondents selected both dubstep and trance. The proportion of students that like either dubstep or trance is:

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which requires meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The probability of getting a Bachelor of Science degree, given that a student is getting a Bachelor of Arts degree, is given by P(Science | Arts) = 0.12. The probability of a student getting both a Bachelor of Arts degree and a Bachelor of Science degree is:

Suppose we toss a coin and roll a die. Let A be the event that the number of spots showing on the die is three or less, and let B be the event that the coin comes up heads. The events A and B are:

Veterinarians suspect that enteroliths (calcifications in the gut of horses) are related to diet-in particular, alfalfa. To investigate this suspicion, a group of veterinarians collected information on the diet of horses and whether the horses developed enteroliths. The table below displays the findings (a case is a horse with enteroliths, a control is without enteroliths). There is a relationship between being fed alfalfa and having enteroliths because:

A group of college DJs surveyed students to find out what music to plan for their upcoming parties. Thirty percent of the students preferred dubstep, 25% of the students liked trance music, and 20% wanted to hear only house music. Fifteen percent of the respondents selected both dubstep and trance. The proportion of students that didn't select any of the music options available on the survey was:

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which requires meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The study also shows that the probability a student gets no degree is P(no) = 0.2. The probability a student gets only a Bachelor of Arts degree is:

In the Virginia instant lottery, there are 10 different $5 Scratcher games. Your favorite, "Hit the Jackpot," is advertised to have a 1-in-4.37 chance of winning and a 1-in-664,457 chance of hitting the top prize of $200,000. All of the seven top prizes are still available. If you buy five of these tickets and outcomes are independent, the probability of winning at least once in these five draws is:

After observing the defects within individual candies in many bags of M&Ms, it has been determined that 11% of all candies are defective, that the probability of observing an M&M with a missing letter is 22%, and that the probability of observing a cracked M&M, given that you already know it is defective, is 70%. You can calculate the probability that you randomly select an M&M that is cracked because:

In the Virginia instant lottery, there are 10 different $5 Scratcher games. Your favorite, "Hit the Jackpot," is advertised to have a 1-in-4.37 chance of winning, and a 1-in-664,457 chance of hitting the top prize of $200,000. All of the seven top prizes are still available. If you buy five of these tickets and outcomes are independent, the probability that you will not win the jackpot is:

A group of freshmen at a local university consider joining the equestrian team. Thirty-five percent of students choose Western riding, 45% choose dressage, and 50% choose jumping. Twenty percent choose both dressage and jumping, while 10% choose Western and dressage. No one chooses Western and jumping. There are no horses suitable for two styles, and each student is assigned to one horse. If two students decide to join the team, what is the probability that both are Western and dressage riders, if they independently decide?

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which require meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The study also shows that the probability a student gets no degree is P(no) = 0.2. Some probability calculations show the probability of getting a Bachelor of Arts and Sciences to be P(Arts & Science) = 0.1. Getting a Bachelor of Arts degree and getting a Bachelor of Science degree are:

Spelling mistakes in a text are either "nonword errors" or "word errors." A nonword error produces a string of letters that is not a word, such as "the" typed as "teh." Word errors produce the wrong word, such as "loose" typed as "lose." Nonword errors make up 25% of all errors. A human proofreader will catch 80% of nonword errors and 50% of word errors. What percent of errors will the proofreader catch?

Let A be the event that a flight from New York to San Francisco arrives on time and let B be the event that it is a clear day in San Francisco. Suppose the probability of a clear day is P(B) = 0.6 and the probability a plane arrives on time is P(A) = 0.7. We also know that the probability a plane arrives on time on a cloudy day is P(A | B^C) = 0.5. The probability a plane arrives on time and it is a cloudy day is:

University degree requirements typically are different for Bachelor of Science degrees and Bachelor of Arts degrees. Some students get a Bachelor of Arts and Science degree, which requires meeting graduation criteria for both degrees. A student advisor needs to know the probability a newly admitted student is interested in such a program, so that the student can be properly advised. A study of previous years finds that the probability a student gets a Bachelor of Science degree is P(Science) = 0.3 and the probability a student gets a Bachelor of Arts degree is P(Arts) = 0.6. The study also shows that the probability a student gets no degree is P(no) = 0.2. The probability a student gets a Bachelor of Arts and Science degree is: