Algebra II : Probability
Study concepts, example questions & explanations for Algebra II
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Example Questions
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 #11 : 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 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?
No. Bob had a higher probability of interviewing the local expert than Ann or Cathy did.
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.
Yes. All three students have equal probabilities of interviewing the expert.
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 #12 : 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 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%
1.0 or 100%
Cannot be determined
0.50 or 50%
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 #4 : 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. 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. 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.
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.
Example Question #1 : Experiment Design And Identifying Bias
A researcher seeks to test the effectiveness of an experimental treatment for strep throat. She randomly divides her study participants, all of whom have been diagnosed with strep throat, into two groups: one group receives the experimental treatment.
Should the other group receive a placebo treatment?
No. Only one group should receive any kind of treatment.
Yes. Placebos help to verify whether an experimental treatment is effective.
No. It is unethical to provide placebo treatments to human participants.
Yes. Providing both groups with a treatment is fair.
Yes. Placebos help to verify whether an experimental treatment is effective.
It is well documented that when people believe they are receiving a treatment, they experience what they expect will be the effects of that treatment. Placebos help to verify whether an experimental treatment is effective by reducing the likelihood that study participants’ expectations or beliefs about treatments, or about whether they received a treatment, skew study results.
Example Question #1 : Experiment Design And Identifying Bias
A researcher seeks to test the effectiveness of an experimental treatment for strep throat. She randomly divides her study participants, all of whom have been diagnosed with strep throat, into two groups: one group receives the experimental treatment, while the other receives a placebo treatment. No study participant is told whether he or she has received the experimental treatment or the placebo, but the researcher keeps careful track of which participants have received the experimental treatment and which have not.
What is the design of this study?
Subjective
Double-blind
Observational
Blind
Blind
It is well documented that when people believe they are receiving a treatment, they experience what they expect will be the effects of that treatment. In a blind study, no participant being studied is aware which group he or she is in, or exactly what treatment he or she received. This can help prevent study participants’ expectations and beliefs about treatments, or about whether they received a treatment, from skewing study results.
Example Question #1 : Experiment Design And Identifying Bias
A researcher seeks to test the effectiveness of an experimental treatment for strep throat. She randomly divides her study participants, all of whom have been diagnosed with strep throat, into two groups: one group receives the experimental treatment, while the other receives a placebo treatment. During the study, neither the researcher nor the study participants know who has received the experimental treatment, and who has received the placebo.
What is the design of this study?
Observational
Blind
Double-blind
Subjective
Double-blind
When researchers are aware which study participants have received which treatments, they may consciously or subconsciously behave in ways that skew results of the study towards the researchers’ own perspectives or interests. In a double-blind study, during the study neither the researchers nor any participant is aware which group he or she is in, or exactly what treatment he or she received. This can prevent researcher behavior from skewing study results.
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