12 min read•november 18, 2021

Jerry Kosoff

The FRQ is a great way to prep for the AP exam! Review FRQ practice writing samples and corresponding feedback from Fiveable teacher Jerry Kosoff!

A high school debate team has 15 members, 10 of whom are “upperclassmen” (11th or 12th grade) and 5 of whom are “lowerclassmen” (9th or 10th grade). For an upcoming tournament, the debate coach wants to use a random-selection process to choose who will attend, as all of the members of the team have expressed interest. There are 4 slots available for the tournament. The debate coach writes every member’s name on an evenly-sized index card, shuffles the index cards, then places them in an envelope before selecting four cards without replacement. Let X represent the number of lowerclassmen selected for the tournament.

a. Is X a binomial random variable? Explain why or why not.

b. Calculate the probability that the four selected team members are all lowerclassmen.

c. After the selection of the four team members resulted in all lowerclassmen, one of the upperclassmen decided to run a simulation to model the probability of that occurring. Here is the proposed simulation design:

Does the proposed simulation design provide an accurate model for simulating the selection process designed by the debate coach? Explain why or why not.

a) X is not a binomial random variable as one outcome of one trial affects the outcome of the next trial. The probability of success is not constant, as after one lowerclassmen is drawn in the first trial with a probability of 5/15, the next trial has a different probability of success of 4/14.

b) P(X = 4) = 5/15 * 4/14 * 3/13 * 2/12 = 0.0036

There is a 0.0036 probability that the four selected team members are all lowerclassmen.

c) This proposed simulation design does provide an accurate model for simulating the selection process designed by the debate coach. This is because it provides an equal probability of success (0.33) every trial with each trial ending after four integers.

Teacher Feedback

Your responses in part (a) and (b) are clear, concise, and demonstrate understanding of the topic at hand (“the probability of success is not constant”) is the key in part (a). In part ( c ), however, you mention that each trial of the simulation would have the same probability of success… which is the exactoppositeof what you mentioned in part (a). So the simulation wouldnotdo what the debate coach proposed, since the simulation would have independent trials; the debate coach did not. [An extension: the simulation would simulate getting P(x = 4) as abinomialrandom variable] Therefore you would not earn credit for part ( c ).

a) X is not a binomial random variable because the trials ran to select a student are not independent. The outcome of a trial can have an effect on the other trial. Also, the parameter § which is the probability of success is not the same as it changes after each trial.

b) P(x=4)= 5/15*4/14*3/13*2/12=0.0037
There is a 0.0037 probability that the four selected team members are all lowerclassmen.

c) No, the proposed simulation design provided is not an accurate model for simulating the selection process designed by the debate coach because the coaches method as mentioned in part (a) has parameter § which is the probability of success is not the same as it changes after each trial. However, the proposed simulation design by one of the upperclassmen would have the probability of success remain the same (5/15 or 0.33).

Teacher Feedback

In part (a), you give the correct reasoning (the trials are not independent), and explain what that represents… in general. That is, your answer is not presented in the context of the scenario. To ensure you get the “context” component, name the variable involved or describe what is being measured in some way. In this example, instead of “the outcome of a trial can have an effect on the other trials”, you could say “the outcome of the first student being picked impacts the results of the other selections” or “the probability of drawing a lowerclassmen would be dependent on the previous selections” or something to that effect.

Your work in part (b) is correct, with correct calculations and work shown. Full credit there.

In part ( c ), you do a good job of defending your answer of “no” - explaining why the proposed simulation results inindependenttrials, whereas the selection process originally proposed does not. This is the correct explanation. Nice job!

a. X is not a binomial random variable. In a binomial distribution, the conditions are needed for independent and probability of success constant. In the question, it states “selecting 4 cards without replacement” meaning that it would change the probability of the events and also being dependent. For example, if one lower class man got pick, the probability of getting another lower-class man decreases to only 4 rather than 5.

b. P(all lower classman)= ( 5/15 * 4/14 * 3/13 *2/12 ) = 0.00366

c. NO, the proposed simulation design provides an accurate model for simulating the selection process designed by the debate coach. It states “Repeated numbers are allowed” meaning one person could be picked twice and that does not make sense.

Teacher Feedback

I’m going to take a stab at explaining “independent” in this context in writing: for trials to be “independent,” the probability of a successful trial (in this case, picking a lowerclassman) must remainconstantfrom one trial to the next (as you mention). So, as you say, the scenario does not lend itself to independent trials, and binomial calculations do not apply. However, when you say “the probability … decreases to only 4 rather than 5”, we should be careful: probability can’t be whole numbers like that (only proportions between 0 and 1), so that is a misuse of the word (we could say 4/14 rather than 5/15 and be OK though). Please let me know if the above doesn’t address your question.

Your answer to part (b) is spot on - work shown and everything. For part ( c ) , you correctly answer “no” while citing the idea that one person could be picked twice. I think you can strengthen your answer by making it clear that that cannot happen when we sample without replacement (instead of the more vague “that does not make sense”). You should still land somewhere near full credit with the response, though.

A. No, X is not a binomial random variable because it doesn’t satisfy the binomial requirements. The probability of success is not the same for each trial because the debate coach is not sampling with replacement.

B. P(X=4)= (5/15)(4/14)(3/13)(2/12)=.0036

C. No, the proposed simulation design does not provide an accurate model for simulating the selection process designed by the debate coach. In the simulation proposed, it states that repeated numbers are allowed, however, when the debate coach determined which 4 slots to give, he selected without replacement thus changing the probability of success. From the very start, the probability of success should have been constant so it results in independent trials.

Teacher Feedback

Good responses! You do a good job of being concise, while still saying everything needed to demonstrate understanding. In part ( c ), you could do a little more to demonstrate that the simulation results inindependenttrials (you describe this without explicitly naming it), which is why the design is not an accurate model.

A. X (the number of lowerclassmen selected for the tournament) is NOT a binomial variable. Due to the fact that the debate coach is selecting four cards WITHOUT replacement makes the number of lowerclassmen dependent, because the probability of a lowerclassman being chosen to attend the tournament changes as the cards are not replaced.
B. P(X=4)= (5/15)(4/14)(3/13)(2/12)=.00366; the probability that the four selected members are all lowerclassmen is .366%.
C. No, this model is not an accurate model for simulating the selection process designed by the debate coach due to the fact that it is independent, meaning that the chance of either an upperclassman or a lowerclassman remains constant, while in the debate coach’s design, this wasn’t the case because the debate coach picked name without replacement.

Teacher Feedback

Nicely done! All three parts are thorough and contain correct information!

a) x is not a binomial random variable since sample without replacement makes the probability of success differ for each trial which does not meet the condition for a binomial of having the same probability of success for each trial
b) P(x=4) = 5/15 x 4/14 x 3/13 x 2/12 = 0.0036
c) Np the proposed simulation is not an accurate design since, as mentioned in part A, in reality, each trial is not independent meaning that the probability of choosing an upperclassmen or underclassmen changes each time- because of the coaches method of picking without replacement

Teacher Feedback

Nice job! One thing to point out: in part ( c ), you defend your answer by mentioning “the probability of choosing an upperclassmen or underclassmen changes each time” - that providescontextin your response. Your answer in part (a), while correct, does not have that same level of context (you just say “does not meet the condition of having the same probability of success for each trial”, which is agenericanswer and not in-context for this problem). Be sure to include context wherever possible in your answers.

a) X is not a random variable because the trials are not independent of each other. One outcome affects the outcome of another trial as the probability of success changes for each lowerclassmen selected without replacement.

b) P(X=4)= 5/15 * 4/14 * 3/13 * 2/12= 0.0036
The probability that the four selected team members are all lowerclassmen is 0.0036.

c) No the simulation design does not provide an accurate model for simulating the selection process by the debate coach. The debate coach designed a random-selection process that did not have trials that were independent of each other. The trials in the simulation design using the random number generator are independent of each other and one outcome doesn’t affect another outcome. This is not the design proposed by the debate coach.

Teacher Feedback

Well done! This looks like a case of all three answers correct with appropriate supporting work/details.

a) No, X (the number of lowerclassmen selected for the tournament) is NOT a binomial random variable. Although it is binary, which means there are exactly 2 outcomes for each trial --> either a lower classman is selected §, or an upperclassman is selected (q), the trials are not independent of each other. Once one student is selected, the probability that the next student is selected is going to change. For example, the probability that a lowerclassmen is selected for the first trial is 5/15, but once a lowerclassmen is selected, the probability will be 4/14. The probability that an upperclassmen/low-classman is selected is going to change because we are selecting students without replacement.

b) P(all of the 4 students are lowerclassmen) 5/15 x 4/14 x 3/13 x 2/12=.0036 The probability that the four selected students are all lowerclassmen is .0036 or .36%.

c) No, upperclassmen’s proposed simulation design does not provide an accurate model for simulated the selection process designed by the debate coach. The coach’s design involved writing each of the student’s name on an envelope. This gave each of the 15 students an equal opportunity of being selected and excluded any repeats. The upperclassmen’s simulation allowed repeated numbers, which potentially result in students being selected more than once. The probability that a lowerclassmen is selected remains constant throughout the experiment (5/15), whereas in the coaches simulation, the probability that a lowerclassmen is selected will change for each trial.

Teacher Feedback

Very good description on part (a)! You give your explanation for the trials not be independent in context of the situation. For part (b), you’ve correctly calculated the probability with appropriate supporting work, and in part ( c ) you make it clear that the proposed simulation has independent trials (you don’t say the word independent, but your description is clear enough that it is understood). Nice job!

a) NO, because in order for it to be a binomial variable the situation must be **B**inary, **I**ndependent, and have a fixed **N**umber of trials. This situation, however, isn’t independent as every time a lowerclassmen is chosen the probability for another lowerclassmen to be chosen changes.
b) p(x=4)=5/15 X 4/14 X 3/13 X 2/12= 1/273 or 0.003663
c)No, since in this simulation repeats are allowed whereas in the coach’s situation “the probability of success is not constant.” The simulation is Independent whereas the actual situation was not independent.

Teacher Feedback

Nice job! You’ve given clear explanations/calculations in all three parts, and all are correct.

a) No, because the index cards are drawn without replacement, X is not independent. A binomial random variable has to be independent.

b) Since X is not independent, the size will reduce by one after each draw.

5/15 * 4/14 * 3/13 * 2/12 = 0.00366 or 0.36%

c) No, the proposed simulation is not an accurate model of the selection process designed by the debate coach because it has replacement. Each time the simulation is repeated, it takes a number from 1-15, but the process designed by the coach reduces the total size by one after each draw of a card. To elaborate, a simulation similar to that of the coach’s would start by generating the first random integer between 1-15, but then it should generate the second random integer between 1-14, the third between 1-13, and the last between 1-12. If the first random integer generated is an upperclassmen (1-10), then the range for upperclassmen for the second generated integer would be from 1-9 and the range for the lowerclassmen would be from 10-14. Same if the first integer was that of a lowerclassmen. This process is repeated until you have 4 random integers.

Teacher Feedback

Solid work. On part (a), you’ll need to do more to explain why X is not independent - the reason is because we’re drawn without replacement, but you’ll need to explain a little more like you did in part (b) or part ( c ) about the probability changing for each trial.

a. No, X isn’t a binomial random variable because the probability of each student being selected goes up after every index card because the problem states that each student is picked without replacement of the card, which also makes the trials dependent on each other.

b. P(4 team members are lowerclassmen)= (5/15)*(4/14)*(3/13)*(2/12)=.0037

There is a .37% chance that the four selected team members are all lowerclassmen.

c. No, this simulation design would not provide an accurate model for simulating the selection process designed by the simulation because while the coach did not replace the index cards after every random selection, the upperclassman did by stating that repeated numbers are allowed. Using repeated numbers is essentially the same as replacing the index cards because we are allowed to choose the same people in the selection.

Teacher Feedback

Solid work throughout here. Be careful in part (a) - you’re giving the correct answer (no), with the correct reason (the selection without replacement makes the trials dependent), but your description of “the probability of each student being select goes up after every index card” is missing the fact that this is assuming those studentsdidn’t already get picked, which you don’t explicitly mention. It could lead to a false conclusion (once a certain student is picked, the probability they get picked again is not 0, and hence went down). You could edit your statement to say “every time a student is not picked, the probability of them being selected goes up”… hence the trials are dependent. In (b) and ( c ) you do everything correctly.

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