Algebra II : Algebra II
Study concepts, example questions & explanations for Algebra II
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Example Questions
Example Question #7 : Using Probability To Make Decisions
An experimental drug is created to reduce the amount of time patients feel sick with the common cold. In clinical trials of people suffering from the common cold, different participants taking the drug experienced symptoms for varying lengths of time. The scientists running the trial rounded each participant’s duration of symptoms to the nearest day, and used this information to develop the following probability distribution:
There were 
If the probability distribution was constructed based on the real durations of participants’ symptoms, the 
Example Question #8 : Using Probability To Make Decisions
An experimental drug is created to reduce the amount of time patients feel sick with the common cold. In clinical trials of people suffering from the common cold, different participants taking the drug experienced symptoms for varying lengths of time. The scientists running the trial rounded each participant’s duration of symptoms to the nearest day, and used this information to develop the following probability distribution:
If the scientists select one of the participants at random, what duration of symptoms can they expect the participant to have experienced?
The expected duration of a randomly selected patient’s symptoms can be calculated using the expected value yielded by this probability distribution. This expected value can be calculated as follows:
Expected value:
Therefore, the most likely duration of symptoms a randomly selected participant would have is 
Example Question #2 : Using Probability To Make Decisions
A fair coin is flipped 9 times, yielding the following results:
Two students are deciding whether to flip this coin a tenth time, in order to decide which one of them will get to keep the five-dollar bill they found on a sidewalk. Would this be a fair method for making this decision?
Cannot be determined
Yes
No
Yes
As long as the coin is truly fair, meaning that there is a 50% chance that it will land on heads and a 50% chance that it will land on tails, the coin will still be fair on the tenth flip. Past results do not affect the odds of a particular result on a subsequent identical event. Therefore, flipping this coin would be a fair way to decide which of the two students will get to keep the five-dollar bill, since each one of them would have a 50% probability of keeping the bill.
Example Question #10 : Using Probability To Make Decisions
Ann, Bob and Cathy are students working together on a group project for school. The project involves three tasks, each of which one of the three students will complete: creating a model, interviewing a local expert, and writing a report. No student has the time to complete more than one task, and all three of them have a strong preference for interviewing the local expert. They decide to find a fair way to randomly distribute the three tasks among themselves.
Which of the following would be a fair method of accomplishing this, allowing all three of them equal odds of completing their preferred task?
Ann selects from the three tasks. Bob then chooses which he would prefer, and the remaining task is assigned to Cathy.
The three tasks are numbered, and the three resulting numbers are written on separate identical pieces of paper and put into a box. Each student takes turns drawing a piece of paper without looking and is assigned the task corresponding to their number.
The local expert chooses one of the three students.
Bob flips a coin. If it lands heads, he creates the model; if tails, he interviews a local expert. After he is assigned a task by this method, Cathy flips a coin. If it lands heads, she writes the report; if tails, she is assigned the other remaining task. Ann is then assigned whichever task is left.
The three tasks are numbered, and the three resulting numbers are written on separate identical pieces of paper and put into a box. Each student takes turns drawing a piece of paper without looking and is assigned the task corresponding to their number.
"Ann selects from the three tasks. Bob then chooses which he would prefer, and the remaining task is assigned to Cathy. " is incorrect because an intentional choice, particularly when biased by a strong preference for a single outcome, is unlikely to be random. Further, were Ann to choose to interview the expert, Bob and Cathy would never have a chance to be assigned that task, significantly advantaging one of the three students over the other two. "Bob flips a coin. If it lands heads, he creates the model; if tails, he interviews a local expert. After he is assigned a task by this method, Cathy flips a coin. If it lands heads, she writes the report; if tails, she is assigned the other remaining task. Ann is then assigned whichever task is left." is incorrect because Bob’s probability of interviewing the expert is 50% in this scenario, a higher probability than either Cathy or Ann would have. "The local expert chooses one of the three students." is incorrect because an intentional choice is unlikely to be random. "The three tasks are numbered, and the three resulting numbers are written on separate identical pieces of paper and put into a box. Each student takes turns drawing a piece of paper without looking and is assigned the task corresponding to their number. " is correct because any one of the three students may draw any one of the three slips, and cannot see which they are drawing, preventing their personal biases or preferences from influencing their selections.
Example Question #5171 : Algebra Ii
Ann, Bob and Cathy are students working together on a group project for school. The project involves three tasks, each of which one of the three students will complete: creating a model, interviewing a local expert, and writing a report. No student has the time to complete more than one task, and all three of them have a strong preference for interviewing the local expert. They decide to randomly distribute the three tasks among themselves in a fair manner, such that all three of them have equal odds of completing their preferred task.
After some consideration, they reach for a coin. Bob flips it; if it lands heads, he creates the model; if tails, he interviews a local expert. After he is assigned a task by this method, Cathy flips a coin. If it lands heads, she writes the report; if tails, she is assigned the other remaining task. Ann is then assigned whichever task is left. By this method, the group decides that Bob will create the model, Cathy will interview the local expert and Ann will write the report.
Was their chosen method fair?
Yes. All three students have equal probabilities of interviewing the expert.
No. Ann never flipped the coin, so she never had an opportunity to obtain a coin result that would assign her to interview the local expert.
Yes. Coins have equal odds of landing on heads or tails, so flipping a coin is a fair way to make a decision between options.
No. Bob had a higher probability of interviewing the local expert than Ann or Cathy did.
No. Bob had a higher probability of interviewing the local expert than Ann or Cathy did.
Bob’s coin flip gives him a 50% chance of obtaining each of the two tasks assigned to his coin flip (creating the model or interviewing the local expert), so he has a 50% chance of interviewing the local expert. The total probability that the task will be assigned to one of the three of them is 1. Ann and Cathy both have a nonzero chance of interviewing the local expert; for instance, if Bob and Cathy both land heads, Ann would be assigned her preferred task. The following relationships express this:
Ann’s probability + Bob’s probability + Cathy’s probability = 1
A + B + C = 1
A + 0.5 + C = 1
A + C = 0.5
C > 0
A > 0
Therefore, both C and A must be less than 0.5, so Bob’s probability of completing his preferred task is higher than both Cathy’s and Ann’s probabilities.
Example Question #5172 : Algebra Ii
Ann, Bob and Cathy are students working together on a group project for school. The project involves three tasks, each of which one of the three students will complete: creating a model, interviewing a local expert, and writing a report. No student has the time to complete more than one task, and all three of them have a strong preference for interviewing the local expert. They decide to randomly distribute the three tasks among themselves in a fair manner, such that all three of them have equal odds of completing their preferred task.
After some consideration, they reach for a coin. Bob flips it; if it lands heads, he creates the model; if tails, he interviews a local expert. After he is assigned a task by this method, Cathy flips a coin. If it lands heads, she writes the report; if tails, she is assigned the other remaining task. Ann is then assigned whichever task is left.
What is the probability that Ann is assigned to interview the expert?
0.25 or 25%
Cannot be determined
0.50 or 50%
1.0 or 100%
0.25 or 25%
Ann’s probability of interviewing the expert is reliant on two events: Bob’s coin toss and Cathy’s coin toss. For Ann to interview the expert, Bob must flip heads and Cathy must then flip heads as well. Each coin toss has a 50% odds of any one outcome, so Ann’s probability of interviewing the expert may be calculated as follows:
Event 1 * Event 2 = Ann’s probability
0.5 * 0.5 = 0.25
Therefore, Ann’s probability of interviewing the expert is 0.25, or 25%. Note that this method of allocating tasks is therefore not fair, as the three students will not have equal probabilities of interviewing the expert.
Example Question #1 : Experiment Design And Identifying Bias
Yes
No
No
No, this experiment is not statistically well designed. In order to be statistically well designed, an experiment's subjects must be randomly sorted into two groups (experimental and control), everyone in the experimental group gets the experiment, and everyone in the control group does not get the experiment. All 80 participants were asked only to participate in the experimental trial; no one was placed into a control group. For this reason, this experiment is not statistically well designed.
Example Question #2 : Experiment Design And Identifying Bias
Krystal owns a boutique clothing store, and wants to learn more about her customers. She sends out a survey to the 35 customers who spent the most money in her store last month. Is this sample of customers likely to be biased?
Yes
No
Yes
Yes, this sample is likely to be biased. If Krystal only reaches out to the customers who spent the most money, she may fail to understand all types of consumer habits.
Example Question #3 : Experiment Design And Identifying Bias
No
Yes
Yes
Example Question #2 : Experiment Design And Identifying Bias
A student wants to find out whether mothers in his city support a local initiative to redesign the city’s largest park. He decides to conduct a poll on this question by calling the home phone numbers of all 200 students in his private high school, asking to speak to their mothers, and recording their responses as YES, NO or NO PREFERENCE in a chart. He doesn’t have time to make 400 calls, so he enlists three friends, who are also students at his school, to help make calls and record answers.
100 calls are not answered, and during 50 calls no mothers are home. The remaining 250 calls successfully reach mothers: 200 of the 250 calls are recorded as YES, 25 are recorded as NO and 25 are recorded as NO PREFERENCE. The student concludes that most mothers in the city support the initiative.
Which of the following best describes whether this study is well-designed, and why?
Yes. The student and her friends called all 400 students in his high school, excluding none.
No. Some calls were not answered.
No. The mothers of students at her school are not likely to be representative of mothers in the city as a population.
Yes. The answers recorded correctly reflect the number of calls made and answered.
No. The mothers of students at her school are not likely to be representative of mothers in the city as a population.
Given the small population of students at the high school, it is highly unlikely that all mothers in the city are represented in the sample; and given that the high school is both small and private, it is unlikely that the sample is representative of all mothers in the city, creating sampling bias.
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