A cooler contains three types of beverages

2018 Handbook

2018 Test B Math · Question 16 of 57

Also in 2026 Practice Test B Math, question 14; 2025 Practice Test B Math, question 16.

Question

A cooler contains three types of beverages: 5 bottles of apple juice, 3 bottles of grape juice, and 6 bottles of orange juice. What is the probability that a bottle chosen at random from this cooler is not apple juice?

A $\frac{1}{9}$
B $\frac{5}{14}$
C $\frac{9}{14}$
D $\frac{2}{3}$

Show the answer

Correct answer: C

The total number of bottles of juice in the cooler is

5 + 3 + 6 = 14

The number of bottles of juice that are not apple juice (grape juice and orange juice) is

3 + 6 = 9

So the probability is $\frac{9}{14}$ .

Question analysis

This probability question describes a cooler holding apple, grape and orange juice bottles in different numbers and asks for the chance that a randomly chosen bottle is not apple juice. Students count favorable bottles over all bottles. The word not is the trap, along with counting kinds of drink instead of bottles.

NYS standards: 6.SP.8a uniform probability models
Grade level: Grade 6 content, by NYS standard
Compared with grade 7: Below grade 7 Simple probability with equally likely outcomes is grade 6 content in New York. This needs one count and one fraction, and the only extra demand is reading the word not, so it sits below grade 7.
Estimated difficulty: Straightforward: two steps or one careful reading.
Thinking it demands: Depth of knowledge 1, recall and reproduction · mainly apply
What makes it hard: Turning words into math minor
Solution: 2 steps
Reading level: Question and choices: about grade 6, 41 words.
Students, last 30 days: Not enough answers yet to show (it takes 10).

Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The counting is easy, 14 bottles in all, so the question mainly tests careful reading. The word “not” turns the event around, and the probability of the opposite event is waiting among the choices. The other wrong choices catch students who count kinds of juice instead of bottles, or who build a fraction with the wrong total.

Turning words into math minor Understanding the question
The event “not apple juice” must be turned into grape plus orange, 3 + 6 bottles over all 14, so the word “not” decides which counts go in the numerator. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

Strategy: Add all the bottles for the denominator, then count only the grape and orange bottles for the numerator, or subtract the chance of apple juice from 1.

Traps

Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".

The wrong answers, classified

A Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: weak
Puts 1 over the 9 bottles that are not apple juice, as if one particular bottle had to be picked.
B Misreads the question (Hegarty, Mayer & Monk, 1995)
Reveals a slip Expected pull: strong
Finds the probability of choosing apple juice, 5 out of 14, missing the word “not” in the question.
D Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Reveals a misconception Expected pull: moderate
Counts two of the three kinds of juice as not apple and treats the kinds as equally likely, ignoring how many bottles each has.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Probability: $P = \frac{\text{outcomes you want}}{\text{all outcomes}}$; P is the probability, the chance that something happens. The top counts the outcomes you want, here bottles that are not apple. The bottom counts all outcomes, here every bottle. It works when every bottle has the same chance.
Probability of “not”: $P(\text{not } A) = 1 - P(A)$; A is an event, such as picking apple juice. P(A) is its chance. Every bottle either is apple or is not, so the two chances add to 1. Subtract from 1 to get the chance that A does not happen.

Sort out the question
The cooler has 5 apple, 3 grape and 6 orange bottles. One bottle is picked at random. That means every bottle has the same chance.
You need the chance that it is not apple juice.
Count all the bottles
$5 + 3 + 6 = 14$
There are 14 bottles. This is the bottom of the probability fraction, since any of them could be picked.
Count the bottles that are not apple
The bottles that are not apple are the grape and orange ones.
$3 + 6 = 9$
There are 9 such bottles. This is the top of the fraction.
Write the probability
$P = \frac{\text{outcomes you want}}{\text{all outcomes}} = \frac{9}{14}$
This fraction cannot be simplified. 9 and 14 share no factor except 1.
Watch out: $\frac{5}{14}$ is the chance of picking apple juice. That is choice B. Read the word “not” carefully.
So the answer is C, $\frac{9}{14}$ .

Check your answer

Use the “not” rule. The chance of apple is $\frac{5}{14}$ . So the chance of not apple is $1 - \frac{5}{14} = \frac{9}{14}$ . Both ways give the same answer.

The original DOE question

Question 73 as printed on page 156 of the DOE handbook. — As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 156, question 73. Open page 156 of the handbook (PDF)