Terri Latham teaches AP Statistics, Algebra 1 and SAT Prep and is the Math Department Chair at Chesapeake High School in Pasadena, Maryland. Terri is an avid reader and enjoys walking and hiking. Terri has written both multiple choice and free response questions for the College Board. Terri has been an AP Stat Reader, Table Leader, Early Table Leader, and Question Leader for a total of 13 years, in 2007, 2008, then from 2014 to the present. Terri was the Question Leader for 2024 #4.
This was an interesting probability question as it used the geometric distribution, which is likely to be removed from the course in the 2026-2027 school year. While probability questions are fun to score since the answers are mostly numeric, students must also show their work clearly, and provide a correct interpretation for the mean.
In an online game, players move through a virtual world collecting geodes, a type of hollow rock. When broken open, these geodes contain crystals of different colors that are useful in the game. A red crystal is the most useful crystal in the game. The color of the crystal in each geode is independent and the probability that a geode contains a red crystal is 0.08.
Part a:
(a) Sarah, a player, will collect and open geodes until a red crystal is found.
(i) Calculate the mean of the distribution of the number of geodes Sarah will open until a red crystal is found. Show your work.
(ii) Calculate the standard deviation of the distribution of the number of geodes Sarah will open until a red crystal is found. Show your work.
WOULD THIS GET CREDIT?
In order to receive full credit in part (a), the response must provide the correct mean (with supporting work) and standard deviation (with supporting work) based on the given distribution. At least 3 out of 4 components are needed to score an essentially correct (E), while exactly 2 components scores a partially correct (P).
Response 1 does not receive credit because it provides the correct formulas but does not have the correct values substituted into the equations, or the correct answers.
Response 2 does not receive credit because it provides an incorrect mean (rounded) and a correct standard deviation, but with no work shown.
Response 3 receives partial credit because it provides the correct answers without supporting work.
Responses 4 and 5 receive full credit for providing the correct answers with supporting work. In Response 4, rounding the mean was not acceptable, but rounding the standard deviation was acceptable since rounding to one decimal place is 12. Since only 3 parts are needed for full credit, response 4 earns full credit.
Teaching tips:
Make sure that students understand not to round means.
Make sure that students know to be careful with calculations, such as square roots.
Make sure that students understand that taking the square root of just the numerator is different than taking the square root of the entire fraction.
Teach students to substitute the correct values in AND write the answer when using formulas. Do not just write the formulas with symbols or the formulas with numbers without the answer.
Part b:
(b) Another player, Conrad, decides to play the game and will stop opening geodes after finding a red crystal or when 4 geodes have been opened, whichever comes first. Let Y = the number of geodes Conrad will open. The table shows the partially completed probability distribution for the random variable Y.
(i) Calculate P(Y = 3). Show your work.
(ii) Calculate P(Y = 4). Show your work.
WOULD THIS GET CREDIT?
In order to receive full credit in part (b), the response must provide the correct probabilities (i and ii) based on the given distribution, with supporting work shown (i and ii). At least 3 out of 4 components are needed to score an essentially correct (E), while exactly 2 components scores a partially correct (P).
Response 1 does not receive credit because only one probability is correct, with no supporting work shown. The answer to part (ii) is the most common incorrect answer, where the probability of exactly 4 is calculated without accounting for stopping at 4 if a red geode is not found.
Response 2 receives partial credit because both answers are correct, but the supporting work is not sufficient. If a student chooses to use calculator notation, all the inputs must be labeled.
Response 3 receives partial credit because both answers are correct, but no supporting work is shown.
Responses 4 and 5 receive full credit because both probabilities are correct with correct supporting work shown. Response 5 shows the preferred supporting work using formulas, but the calculator notation in Response 4 is acceptable since the inputs are correctly labeled.
Teaching tips:
Teach your students to always answer the question that was asked. I know that this sounds silly, but there was a substantial number of students who found the incorrect probability for part (ii) and didn’t realize that it was incorrect. Make sure to model checking your answer and making sure it answers the question.
If students are going to use calculator function syntax, they must identify (label) the parameters. Although some readers get extremely upset over this because “we know what they mean,” this is the standard for AP Statistics scoring guidelines.
When using newer calculators, like the NumWorks calculator, make sure that students show everything that is on the screen including the name of the distribution and a sketch of the distribution that they chose (which is equivalent to choosing pdf or cdf on a TI calculator).
Part c:
(c) Consider the table and your results from part (b).
(i) Calculate the mean of the distribution of the number of geodes Conrad will open. Show your work
(ii) Interpret the mean of the distribution of the number of geodes Conrad will open, which was calculated in part (c-i).
WOULD THIS GET CREDIT?
In order to receive full credit in part (c), the response must provide the correct (or consistent with part b) mean with appropriate supporting work and the correct interpretation with the following three components: (1) the concept of repeating the selection process many, many times, (2) the concept of an average or mean, and (3) the context of the number of geodes opened.
(4 out of 5 parts needed to score E, must have correct mean with work)
(2 or 3 of mean, work, long run and mean concept for a P)
Response 1 does not receive credit because it only has the correct mean and the concept of an average or mean. The supporting work is not sufficient, there is no concept of repeating the process many, many times and there is no context.
Response 2 does not receive credit because it has only the concept of an average or mean and the context of number of geodes opened. Although the mean looks correct, it is the result of an incorrect process and therefore is not acceptable.
Response 3 receives partial credit. The mean and supporting work are acceptable, but there is no concept of repeating the process many, many times, no concept of mean and the context is incorrect. (opening geodes, not number of red geodes)
Responses 4 gets full credit. It has the correct mean and supporting work, assuming that 0.0677 and 0.0623 were their answers in part (b). Response 4 is missing the concept of repeating the process many, many times, but is still scored essentially correct.
Response 5 is the model solution and earns full credit.
Teaching tips:
Give students opportunities to practice interpreting means in context.
Teach your students to reflect on their answers to see if the answer is valid, such as a probability distribution that does not add to one. Although means calculated from an incorrect probability distribution were accepted on this problem, students should realize that the probability distribution was not valid, making part (b)(ii) incorrect.
Make sure students understand that not all decimals are probabilities. When a number is between 0 and 1, it can still be an expected value or mean. If students understand the context, this misunderstanding should be less frequent.
Give students opportunities to interpret numerical values, such as means and standard deviations, not just statistical tests and intervals.
Encourage your students to go back and fix their mistake if they realize that they made one. Some students fixed the probability distribution in part (c) without going back and fixing it in part (b).