Two-step sample-space tables with and without replacement practice questions | Year 10 Mathematics

First / second	1	2	5	3	4
5	1	1	1	1	1
2	1	1	1	1	1
1	1	1	1	1	1
3	1	?	1	1	1
4	1	1	1	1	?

First / second	1	2	5	3	4
5	(5,1)	(5,2)	(5,5)	(5,3)	(5,4)
2	(2,1)	(2,2)	×	(2,3)	(2,4)
1	(1,1)	(1,2)	(1,5)	(1,3)	(1,4)
3	(3,1)	(3,2)	(3,5)	(3,3)	(3,4)
4	(4,1)	(4,2)	(4,5)	(4,3)	(4,4)

First / second	1	2	5	3	4
5	(5,1)	(5,2)	×	(5,3)	(5,4)
2	(2,1)	×	(2,5)	(2,3)	(2,4)
1	×	(1,2)	(1,5)	(1,3)	(1,4)
3	(3,1)	(3,2)	(3,5)	×	(3,4)
4	(4,1)	(4,2)	(4,5)	(4,3)	×

First / second	1	2	5	3	4
5	(5,1)	(5,2)	(5,5)	(5,3)	(5,4)
2	(2,1)	(2,2)	(2,5)	(2,3)	(2,4)
1	(1,1)	(1,2)	(1,5)	(1,3)	(1,4)
3	(3,1)	(3,2)	(3,5)	(3,3)	(3,4)
4	(4,1)	(4,2)	(4,5)	(4,3)	(4,4)

First / second	1	2	5	3	4
5	{1,5}	{2,5}	{5,5}	{3,5}	{4,5}
2	{1,2}	{2,2}	{2,5}	{2,3}	{2,4}
1	{1,1}	{1,2}	{1,5}	{1,3}	{1,4}
3	{1,3}	{2,3}	{3,5}	{3,3}	{3,4}
4	{1,4}	{2,4}	{4,5}	{3,4}	{4,4}

Second / first	4	3	2	6	1	5
6	?	1	1	0	1	1
5	?	1	1	1	1	0
1	1	1	1	1	0	1
2	1	1	0	1	1	1
3	1	0	1	1	?	1
4	0	1	1	1	1	1

Second / first	4	3	2	6	1	5
6	(4,6)	(3,6)	(2,6)	×	(1,6)	(5,6)
5	×	(3,5)	(2,5)	(6,5)	(1,5)	×
1	(4,1)	(3,1)	(2,1)	(6,1)	×	(5,1)
2	(4,2)	(3,2)	×	(6,2)	(1,2)	(5,2)
3	(4,3)	×	(2,3)	(6,3)	(1,3)	(5,3)
4	×	(3,4)	(2,4)	(6,4)	(1,4)	(5,4)

Second / first	4	3	2	6	1	5
6	(4,6)	(3,6)	(2,6)	(6,6)	(1,6)	(5,6)
5	(4,5)	(3,5)	(2,5)	(6,5)	(1,5)	(5,5)
1	(4,1)	(3,1)	(2,1)	(6,1)	(1,1)	(5,1)
2	(4,2)	(3,2)	(2,2)	(6,2)	(1,2)	(5,2)
3	(4,3)	(3,3)	(2,3)	(6,3)	(1,3)	(5,3)
4	(4,4)	(3,4)	(2,4)	(6,4)	(1,4)	(5,4)

Second / first	4	3	2	6	1	5
6	{4,6}	{3,6}	{2,6}	×	{1,6}	{5,6}
5	{4,5}	{3,5}	{2,5}	{5,6}	{1,5}	×
1	{1,4}	{1,3}	{1,2}	{1,6}	×	{1,5}
2	{2,4}	{2,3}	×	{2,6}	{1,2}	{2,5}
3	{3,4}	×	{2,3}	{3,6}	{1,3}	{3,5}
4	×	{3,4}	{2,4}	{4,6}	{1,4}	{4,5}

Second / first	4	3	2	6	1	5
6	(4,6)	(3,6)	(2,6)	×	(1,6)	(5,6)
5	(4,5)	(3,5)	(2,5)	(6,5)	(1,5)	×
1	(4,1)	(3,1)	(2,1)	(6,1)	×	(5,1)
2	(4,2)	(3,2)	×	(6,2)	(1,2)	(5,2)
3	(4,3)	×	(2,3)	(6,3)	(1,3)	(5,3)
4	×	(3,4)	(2,4)	(6,4)	(1,4)	(5,4)

First / second	5	3	6	2	4	1
1	1	1	1	1	1	0
4	1	1	1	1	?	1
2	1	1	1	0	1	?
5	0	1	1	1	?	1
6	1	?	0	1	1	1
3	1	0	1	1	1	1

First / second	5	3	6	2	4	1
1	(1,5)	(1,3)	(1,6)	(1,2)	(1,4)	×
4	(4,5)	(4,3)	(4,6)	(4,2)	×	(4,1)
2	(2,5)	(2,3)	(2,6)	×	(2,4)	(2,1)
5	×	(5,3)	(5,6)	(5,2)	(5,4)	(5,1)
6	(6,5)	(6,3)	×	(6,2)	(6,4)	(6,1)
3	(3,5)	×	(3,6)	(3,2)	(3,4)	(3,1)

First / second	5	3	6	2	4	1
1	(1,5)	(1,3)	(1,6)	(1,2)	(1,4)	×
4	(4,5)	(4,3)	(4,6)	(4,2)	×	(4,1)
2	(2,5)	(2,3)	(2,6)	×	(2,4)	(2,1)
5	×	(5,3)	(5,6)	(5,2)	(5,4)	(5,1)
6	(6,5)	(6,3)	×	(6,2)	(6,4)	(6,1)
3	(3,5)	×	×	(3,2)	(3,4)	(3,1)

First / second	5	3	6	2	4	1
1	(1,5)	(1,3)	(1,6)	(1,2)	(1,4)	(1,1)
4	(4,5)	(4,3)	(4,6)	(4,2)	(4,4)	(4,1)
2	(2,5)	(2,3)	(2,6)	(2,2)	(2,4)	(2,1)
5	(5,5)	(5,3)	(5,6)	(5,2)	(5,4)	(5,1)
6	(6,5)	(6,3)	(6,6)	(6,2)	(6,4)	(6,1)
3	(3,5)	(3,3)	(3,6)	(3,2)	(3,4)	(3,1)

First / second	5	3	6	2	4	1
1	{1,5}	{1,3}	{1,6}	{1,2}	{1,4}	×
4	{4,5}	{3,4}	{4,6}	{2,4}	×	{1,4}
2	{2,5}	{2,3}	{2,6}	×	{2,4}	{1,2}
5	×	{3,5}	{5,6}	{2,5}	{4,5}	{1,5}
6	{5,6}	{3,6}	×	{2,6}	{4,6}	{1,6}
3	{3,5}	×	{3,6}	{2,3}	{3,4}	{1,3}

Second / first	2	1	3
1	1	0	1
3	1	1	0
2	0	1	1

Second / first	2	1	3
1	(2,1)	×	(3,1)
3	(2,3)	(1,3)	×
2	×	(1,2)	(3,2)

Second / first	2	1	3
1	(2,1)	(1,1)	(3,1)
3	(2,3)	(1,3)	(3,3)
2	(2,2)	(1,2)	(3,2)

Second / first	2	1	3
1	{1,2}	×	{1,3}
3	{2,3}	{1,3}	×
2	×	{1,2}	{2,3}

Second / first	2	1	3
1	(2,1)	×	×
3	(2,3)	(1,3)	×
2	×	(1,2)	(3,2)

Two-step sample-space tables

Track individual objects and their order before counting an event

Identify physical objects Number the individual counters. Equal colours do not make two counters the same object
Keep the stage order The first and second entries record different draws \(\displaystyle (1,3)\ne(3,1)\)
Apply replacement Returning an object makes all n objects available again. Keeping it out leaves n−1 choices \(\displaystyle \begin{gathered}N_{\text{return}}=n^2\\N_{\text{keep out}}=n(n-1)\end{gathered}\)
Check the headings A cell is impossible without replacement when its original row and column object numbers match. Shuffling the headings can move these cells away from the visual diagonal
Count equally likely pairs Uniform selection from available individual objects makes every legal ordered pair equally likely \(\displaystyle \Pr(E)=\frac{F}{N}\)
Separate exactly one from at least one Exactly one red excludes two reds. At least one includes one red and two reds

Worked example

A bag has two red and two blue counters. Draw twice uniformly without replacement. Find the probability of at least one red

N=4\times3

\(\displaystyle N=12\)Count ordered pairs of physical counters
\(\displaystyle F_{BB}=2\times1=2\)Either blue may be first, leaving one blue
\(\displaystyle F=12-2=10\)Every pair except both blue has at least one red
\(\displaystyle \Pr(E)=\frac{10}{12}=\frac56\)The twelve individual-object pairs are equally likely

Common mistakes

Equally likely colour labels

✕\(\displaystyle \Pr(RR)=\frac14\)✓\(\displaystyle \Pr(RR)=\frac{2}{12}=\frac16\)

For two red and two blue without replacement, RR has fewer physical pairs than RB

Removing a whole colour

✕\(\displaystyle N=4\times2\)✓\(\displaystyle N=4\times3\)

Remove only the drawn counter, not every counter of that colour

Ignoring order

✕\(\displaystyle (1,2)=(2,1)\)✓\(\displaystyle (1,2)\ne(2,1)\)

These are different first and second draws

Difficulty levels

introRecognise complete ordered tables with replacement

standardApply no replacement and read either table orientation

challengeCount events with repeated labels on distinct objects

examCompare tables and solve colour or digit events under either replacement rule

Questions about this topic

Can the same colour occur twice without replacement?+

Yes, if at least two distinct counters have that colour

Why number counters?+

Numbers keep equally likely physical outcomes separate, even when colour labels repeat

Is the impossible region always the diagonal?+

Only when both axes list the objects in the same order. Compare object identities, not display positions

Does independence mean equal likelihood?+

No. Independence describes whether one result changes another probability; equal likelihood compares the probabilities of outcomes

How do I read a reversed table?+

Use the stage labels on the axes, then write the first-stage object before the second-stage object

What if objects are not selected uniformly?+

Counting cells alone is not enough. Assign each outcome its actual probability and add the favourable probabilities

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 10 Mathematics, chapter 4: Probability

In this chapter

Describe and order probability values Single-step theoretical probability and complements 4A Probability scale and experiments: Experimental probability from observed frequencies Finite events, probability assignments and complements 4A Probability scale and experiments: Design and interpret a probability simulation 4B Venn diagrams and sets: Venn regions, set language and probability Compound events, the addition rule and mutual exclusivity Complete and use a two-way probability table Construct and complete two-way frequency tables Convert between Venn diagrams and two-way tables Conditional probability from exact representations 4D Conditional probability: Conditional probability on Venn diagrams Two-step sample-space tables with and without replacement Represent a multi-stage finite sample space Tree-diagram probability for two-stage experiments Multi-stage probability, multiplication and total probability 4G Independence and simulations: Independence on Venn diagrams Test exact independence 4G Independence and simulations: Design and interpret a probability simulation 4G Independence and simulations: Experimental probability from observed frequencies

Search

Two-step sample-space tables with and without replacement

Read identities, not visual positions