Four slips of paper numbered 1, 2, 3, and 4 were placed in a bag: One slip was selected from the bag, and the number recorded. Without replacement, a second slip was chosen. This process continued until no more slips remained in the bag. One part of a tree diagram representing the sample space is shown. Which number should be placed in the box to complete this part of the tree diagram?

Four slips of paper

2023 Handbook

2023 Test B Math · Question 47 of 57

Question

Four slips of paper numbered 1, 2, 3, and 4 were placed in a bag:

One slip was selected from the bag, and the number recorded.
Without replacement, a second slip was chosen.
This process continued until no more slips remained in the bag.
One part of a tree diagram representing the sample space is shown.

$A partial tree diagram. The top number, 1, branches down to 2, 3, and 4. Below 2 are 3 and 4, below 3 are 2 and 4, and below 4 are 2 and 3. The 4 under the 2 has one branch leading down to an empty box.$

Which number should be placed in the box to complete this part of the tree diagram?

Show the answer

Correct answer: C

The “without replacement” statement means that the bag contained four slips for the first draw, three slips for the second draw, two slips for the third draw, and one slip for the last draw. Following the branch from the top and navigating down to the box, the 1 was chosen first, the 2 was chosen second, and the 4 was chosen third. The only remaining number is 3. The element would be 1243.

Question analysis

Four slips numbered 1 to 4 are drawn from a bag one at a time without replacement, and part of a tree diagram of the outcomes is shown. Students decide which number belongs in the empty box at the end of one branch. It tests reading a tree diagram and understanding what drawing without replacement means.

NYS standards: 7.SP.8b sample spaces for compound events
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: Below grade 7 Tree diagrams for compound events are a grade 7 topic, but here the tree is already drawn and only one box is empty. Following a single branch to that box is shorter than typical grade 7 work with sample spaces.
Estimated difficulty: Routine: one familiar step, little to misread.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly understand
What makes it hard: Reading a figure, table or graph minor
Solution: 2 steps
Reading level: Question and choices: about grade 5, 68 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

Only one branch matters, and once it is read as the slips drawn so far, 1, 2 and 4, the last slip left in the bag is the only one missing. Errors come from not counting the top 1 as a draw, from forgetting that a drawn slip stays out, or from copying a nearby number in the diagram instead of following the path.

Reading a figure, table or graph minor Understanding the question
The box hangs below the branch 1, then 2, then 4, and students must read that path from the top as the order of the first three draws. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

Strategy: Follow the branch from the top down to the box, list the slips already drawn on that path, and name the one still left in the bag.

Traps

Without replacement: In a second pick there is one fewer item, which changes the probability.
Misreads the figure: Reads a value, label or tick spacing in a figure, table or graph wrongly.

The wrong answers, classified

A Misreads the figure (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Reveals partial knowledge Expected pull: moderate
Treats the 1 at the top as the start of the diagram rather than the first slip drawn, so 1 seems still to be in the bag.
B Without replacement (Fischbein & Schnarch, 1997)
Reveals a misconception Expected pull: weak
Lets slip 2, already drawn second on this branch, come up again, missing that a slip drawn without replacement stays out of the bag.
D Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Reveals a surface strategy Expected pull: weak
Copies 4, the number directly above the box, reading the diagram as a pattern of numbers rather than as the order of the draws.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Sort out the question
The bag holds four slips: 1, 2, 3 and 4. Slips are drawn one at a time. “Without replacement” means a drawn slip does not go back in the bag. So each number can be drawn only once.
Read the tree diagram
A tree diagram shows the orders the draws can happen in. Each row going down is one draw. The 1 at the top is the first slip drawn. The numbers below it are the choices for the second draw, and so on.
Follow the path to the box
Start at the top and go down the lines to the box.
- First draw: 1
- Second draw: 2
- Third draw: 4
- Fourth draw: the box
Slips 1, 2 and 4 are already out of the bag.
Find the slip that is left
The bag started with 1, 2, 3 and 4. Only slip 3 has not been drawn. So it must be the fourth draw.
Watch out: The 1 at the top counts as a draw, so 1 is not left. That trap is choice A. Slip 2 was already drawn, so choice B can’t be right. And 4 is just the number above the box, choice D.
So the answer is C, 3.

Check your answer

Read the whole path: 1, 2, 4, 3. Each number from 1 to 4 shows up exactly once. That is just what should happen when no slip goes back in the bag.

The original DOE question

Question 104 as printed on page 204 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 204, question 104. Open page 204 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).