June Beighley teaches AP Statistics, Math, and AP Computer Science at Pittsburgh CAPA 6-12, the Creative and Performing Arts magnet school in Pittsburgh, PA. She is an AP Stats Reader and Table Leader, and she enjoys sharing her love of AP Stats with her students and colleagues. June was an Early Table Leader for 2023 #3.

Ah, probability! Like many Math Teachers turned Stats Teachers, I really like probability, so I was excited to score the probability question at the AP Reading this year. This question assesses the concepts of random variables, discrete probability distributions, conditional probability, and expected value with the fun context of bath fizzies. The answers are mostly numeric, but students must also show work and provide interpretations.

*Bath fizzies are mineral tablets that dissolve and create bubbles when added to bathwater. In order to increase sales, the Fizzy Bath Company has produced a new line of bath fizzies that have a cash prize in every bath fizzy. Let the random variable, X, represent the dollar value of the cash prize in a bath fizzy. The probability distribution of X is shown in the table.*

**Part (a)**

*Based on the probability distribution of X, answer the following. Show your work.*

(i) *Calculate the proportion of bath fizzies that contain $1.*

(ii)* Calculate the proportion of bath fizzies that contain at least $10.*

**WOULD THIS GET CREDIT?**

**Response 1: **

(i) 0.68

(ii) 0.12

**Response 2:**

(i) 1 - (0.2 + 0.05 + 0.05 + 0.01 + 0.01) = 0.68

(ii) 0.68 + 0.2 + 0.05 = 0.93

**Response 3:**

(i) 0.2 + 0.05 + 0.05 + 0.01 + 0.01 = 0.32

1 - 0.32 = 0.68

(ii) 0.05 + 0.05 + 0.01 + 0.01 = 0.12

**Response 4:**

(i) *P*(*X* = 1) = 1 - (0.2 + 0.05 + 0.05 + 0.01 + 0.01) = 0.68

(ii) *P*(*X* ≥10) = 0.05 + 0.05 + 0.01 + 0.01 = 0.12

In order to get full credit in part (a), the response must provide correct numerical answers of (i) 0.68 and (ii) 0.12, with correct supporting work. (Minor arithmetic or transcription errors are not penalized if work is shown.)

**Response 1** is given partial credit because no work is shown for either part, although both numerical answers are correct.

**Response 2** is given partial credit because it provides a correct answer with supporting work in part (i) but in part (ii) calculates the proportion of bath fizzies that contain "at most $10."

**Responses 3 and 4** both receive full credit for providing correct answers with supporting work. The use of probability notation (as seen in response 4) was not required, but it demonstrates good statistical communication.

### Teaching Tips:

Require students to always show their work! Although they may use a calculator on the AP Stats exam, showing work demonstrates good statistical communication. “Bald” answers without supporting work rarely get credit on the AP exam.

Provide plenty of practice with probability terms such as exactly 10, at least 10, at most 10, etc. so that students can distinguish among these events.

**Part (b)**

*Based on the probability distribution of X, calculate the probability that a randomly selected bath fizzy contains $100, given that it contains at least $10. Show your work.*

**WOULD THIS GET CREDIT?**

**Response 1: **

**Response 2:**

**Response 3:**

**Response 4:**

In order to get full credit in part (b), the response must provide the correct conditional probability (0.083) with correct supporting work. (Minor arithmetic or transcription errors are not penalized.)

**Response 1** does not receive credit because it provides an incorrect answer with incorrect supporting work. It shows the general conditional probability formula without defining events A and B. It also shows a misapplication of the formula by multiplying the probabilities of the events “exactly $100” and “at least $10” in the numerator, incorrectly assuming that the events are independent.

**Response 2** is given partial credit because it provides the correct answer without supporting work. The use of specific probability notation, while not required, demonstrates good statistical communication and may be considered a positive for holistic scoring.

**Responses 3 and 4** are both given full credit for providing the correct answer with correct supporting work. The use of specific probability notation (as seen in response 3) was not required, but it demonstrates good statistical communication.

### Teaching Tips:

Emphasize the meaning of conditional probability, and practice doing the calculations with discrete probability distributions as well as two-way tables – see this lesson.

Students will have access to the formula sheet during the AP Exam, so make sure that they understand how the conditional probability formula works. They should know how to obtain the probability of the intersection of events (

*A*⋂*B***)**from a table. For some probability strategies to try before using a formula, read this blog post.Require your students to use correct probability notation with the specific events clearly defined or written out. For example, writing

*P*(*X*= $100 |*X*≥ $10) shows an understanding of the random variable and events and is stronger than writing*P*(100 | ≥ 10) or the generic*P*(*A*|*B*). Although it has not always been required in the rubrics, using correct specific probability notation demonstrates good communication and may be considered a positive when grading holistically.

**Part (c)**

*(c) Based on the probability distribution of X, calculate and interpret the expected value of the distribution of the cash prize in the bath fizzies. Show your work.*

**WOULD THIS GET CREDIT?**

**Response 1**:

L₁= cash prize, L₂= probability

1-VAR Stats(L₁, L₂)

μ = $4.68 cash prize

**Response 2:**

*μ*(*X*) = 1(0.68) + 5(0.2) + 10(0.05) + 20(0.05) + 50(0.01) + 100(0.01) = $4.68

**Responses 3:**

*E*(*X*) = .68(1) + .2(5) + .05(10) + .05(20) + .01(50) + .01(100) = 4.68

The expected value of the distribution of cash prize in the bath fizzies is $4.68.

** Response 4:**

1(0.68) + 5(0.2) + ⋯ = $4.68

The expected value of the average bath fizzy is $4.68.

** Response 5:**

1(0.68) + 5(0.2) + 10(0.05) + 20(0.05) + 50(0.01) + 100(0.01) = $4.18

After many, many trials, the expected cash prize in fizzies is $4.18 on average.

In order to get full credit in part (c), the response must provide the correct expected value of $4.68 with supporting work. (Minor arithmetic or transcription errors are not penalized.) The interpretation must be in context and include the concepts of a mean or average cash prize and of repeating the selection process over a long period of time.

**Response 1** does not receive credit. The response provided the correct expected value but does not include sufficient work. Describing calculator functions such as 1-VAR STATS(L₁, L₂) does not communicate understanding of expected value calculation. Also, the response does not provide an interpretation of expected value. Although the response is scored incorrect, components (1) and (5) were satisfied (correct answer with context), so this may be considered a positive in holistic scoring.

**Response 2** receives partial credit because it provides the correct expected value with correct work, but it does not provide an interpretation of expected value.

**Response 3** receives partial credit because it provides the correct expected value with supporting work. The interpretation sounds like expected value, but it is mostly just copying words from the prompt without including the concepts of average and long-term repeated process.

**Response 4** receives partial credit because the numerical answer is correct, and appropriate work is shown. The interpretation describes an “average bath fizzy,” demonstrating a misunderstanding that the random variable is the “cash prize.”

**Response 5** receives full credit because it correctly interprets the expected value in context, including both the concepts of a mean cash prize and long-term selection process. Work is shown, so the incorrect value of 4.18 is considered a minor transcription error resulting from omitting a term when using a calculator.

### Teaching Tips:

Require students to show their work instead of relying on “calculator speak,” which usually does not receive credit on the AP Stats Exam. In an expected value calculation, have students show a few terms to establish the pattern, followed by ellipses (…) to indicate missing terms. Transcription errors are more evident, and more likely to be forgiven, when the work is written out.

When interpreting an expected value, have students include (1) the concept of average or mean, (2) the concept of repeated trials, and (3) context – see this lesson and the ultimate interpretations guide.

Don’t leave off the decimals! An expected value is an average, so decimal answers are appropriate. Make sure your students don’t automatically round to the nearest integer.

**Part (d)**

*(d) The Fizzy Bath Company would like to sell the bath fizzies in France, where the currency is euros. Suppose the conversion rate for dollars to euros is 1 dollar = 0.89 euros. Using your expected value from part (c), calculate the expected value, in euros, of the distribution of the cash prize in the bath fizzies. Show your work.*

**WOULD THIS GET CREDIT?**

**Response 1: **

**Response 2:**

*x* = 4.1652

**Response 3:**

The expected value of the distribution of the cash prize in euros is 4.17.

**Response 4:**

μ = (0.89)(0.68) + (4.45)(0.2) + (8.9)(0.05) + (17.8)(0.05) + (44.5)(0.01) + (89)(0.01) = 4.1652 euros

**Response 5:**

(0.89)(4.68) = €4.17

In order to get full credit in part (d), the response must include the correct expected value of 4.17, with supporting work and units (€ or “euros”).

**Response 1** does not receive credit because it shows an incorrect answer obtained by dividing by the conversion factor.

**Response 2** receives partial credit because it does not include units. Use of a proportion is an appropriate way to show work in this problem.

**Response 3** receives partial credit because it does not show work but does provide the correct answer with units.

**Response 4** receives full credit because it shows work by first converting prize amounts from dollars to euros, then calculating the correct expected value and providing units. It is not necessary to convert to euros first since multiplying all values in the distribution by a constant (0.89) results in the mean being multiplied by the same constant.

**Response 5** receives full credit because it provides the correct answer with work and units.

### Teaching Tips:

Make sure that students understand the effects of adding or subtracting a constant or multiplying or dividing by a constant on the probability distribution of a random variable. Knowing how the mean and standard deviation are affected can save a lot of tedious calculations – see this lesson.

Require your students to include measurement units in their answers. Encourage the habit of writing the answer in a complete sentence to demonstrate good statistical communication.

After adding up all of the Es, Ps, and Is, some student responses will fall between scores (for example, 2.5 points). In those cases, turn to holistic scoring to decide whether to score up or down. When scoring holistically, review the entire response and look for things that add to or take away from the overall strength of the response and quality of communication.

In this problem, using a correct specific probability statement in part (b) shows good communication (see Responses 2 and 3) and is considered a positive. In part (c) a correct answer in context gets an I, but it shows some understanding of how to find expected value (see Response 1), so it is considered a positive. With these criteria, a response scoring EEIP could be marked up from 2.5 to 3. On the other hand, you would likely score down a response that is missing a major concept such as omitting the expected value interpretation (see part (c) Response 2), so a response scoring EIPE could be marked down from a 2.5 to a 2.

This was a fun problem to score, and it was nice to read a question with mostly numbers for a change! I plan to use this problem during class to practice and review. It will be a good problem to use to remind students to show their work, calculate conditional probability without the formula, and properly interpret expected value.

And in case you’re wondering, bath fizzies with cash prizes are a real thing! Do a quick internet search for “bath bombs cash inside.” Good luck!