Conditional Probability Flashcards: Two-Way Table Practice

Question 1

In P(A | B), which event is the given condition?

Accepted Answer

B. Restrict attention to outcomes in B; A is the event whose probability is requested.

Question 2

When one item is selected uniformly from a finite collection, what count ratio gives P(A | B)?

Accepted Answer

Count in both A and B / count in B, provided the B count is positive. In probability notation, P(A | B) = P(A and B) / P(B), with P(B) > 0.

Question 3

What does the event “A and B” include?

Accepted Answer

Only outcomes that satisfy both A and B. It is their overlap, also written A ∩ B; it need not include every outcome in B.

Question 4

For a complete two-way count table, how many cells should each item contribute to?

Accepted Answer

One cell. Each variable’s categories must be mutually exclusive and exhaustive, so no item is duplicated or omitted.

Question 5

Choose one token uniformly from this complete table. Each item appears once.

Dot No dot
Blue        6      9
Not blue    4     11

What is P(dot | blue)?

Accepted Answer

2/5. The blue group has 6 + 9 = 15 tokens; 6 have dots. Use 6/15 = 2/5.

Question 6

Choose one ticket uniformly from this complete table. Each item appears once.

Stamp No stamp
Striped      9        6
Plain        3       12

What is P(striped | stamped)?

Accepted Answer

3/4. There are 9 + 3 = 12 stamped tickets; 9 are striped. Use 9/12 = 3/4.

Question 7

Choose one paper flower uniformly from this complete table. Each item appears once.

Folded Unfolded
Red           5        7
Not red      10        8

A learner writes P(folded | red) = 5/15. What denominator should replace 15?

Accepted Answer

12. The condition is red, so use the red row total, 5 + 7. The 15 counts folded flowers instead.

Question 8

A complete batch has 40 silver and 20 non-silver badges. Exactly 3/5 of the silver badges are numbered; all other silver badges are unnumbered. How many badges are both silver and numbered?

Accepted Answer

24 badges. Apply the within-silver rate to the silver total: (3/5) × 40 = 24.

Question 9

Choose one book uniformly from this complete table. Each item appears once.

Borrowed On shelf
Hardback          8       10
Paperback        15        7

What is P(borrowed | hardback)?

Accepted Answer

4/9. The hardback group contains 8 + 10 = 18 books; 8 are borrowed. Use 8/18 = 4/9.

Question 10

In a complete registration list, P(completed | online) = 3/4 under uniform selection. What does 3/4 describe in context?

Accepted Answer

Three quarters of the online registrations are completed. The fraction describes the online group, not all registrations or the group of completed registrations.

Question 11

Choose one tile uniformly from this complete table. Each item appears once.

Marked Unmarked
Round           7        3
Not round       5        9

What is P(round | marked)?

Accepted Answer

7/12. The marked column has 7 + 5 = 12 tiles; 7 are round.

Question 12

Choose one token uniformly from this complete table. Each item appears once.

Dot No dot
Blue        6      9
Not blue    4     11

What is P(blue | dot)?

Accepted Answer

3/5. The dotted group has 6 + 4 = 10 tokens; 6 are blue. Use 6/10 = 3/5.

Question 13

Choose one folder uniformly from this complete table. Each item appears once.

Tagged Untagged
New       4        6
Old       8       12

A learner calls 4/10 the probability of “new and tagged.” What fraction answers that joint-probability question?

Accepted Answer

2/15. “New and tagged” means 4 folders out of all 30: 4/30 = 2/15. The 4/10 ratio conditions on new.

Question 14

Choose one bead uniformly from this complete table. Each item appears once.

Holed Solid
Green          0     8
Not green      5     7

What is P(holed | green)?

Accepted Answer

0. The green group is nonempty: 0 + 8 = 8. No green bead is holed, so the ratio is 0/8 = 0.

Question 15

Choose one cup uniformly from this complete table. Each item appears once.

Handle No handle
Blue           9         6
Not blue       3        12

What is P(no handle | blue)?

Accepted Answer

2/5. Of the 15 blue cups, 6 have no handle. Use 6/15 = 2/5; the condition stays blue.

Question 16

In a complete collection of envelopes, 18 are both small and sealed. Exactly 3/4 of all small envelopes are sealed. How many small envelopes are there?

Accepted Answer

24 envelopes. If n is the small-group total, (3/4) × n = 18, so n = 18 ÷ (3/4) = 24.

Question 17

Choose one button uniformly from this complete table. Each item appears once.

Hole No hole
Green        6       4
Not green    4      12

What is P(hole | green)?

Accepted Answer

3/5. The green row has 6 + 4 = 10 buttons; 6 have holes. Use 6/10 = 3/5.

Question 18

A complete note collection gives P(blue | sealed) = 2/7 under uniform selection. Which group is the denominator of this fraction?

Accepted Answer

All sealed notes. The numerator counts notes that are both blue and sealed.

Question 19

A complete collection has 40 large and 30 small labels. Exactly 1/4 of the large labels are stamped; all other large labels are unstamped. How many large labels are unstamped?

Accepted Answer

30 labels. There are (1/4) × 40 = 10 stamped large labels; 40 − 10 = 30 are unstamped.

Question 20

Choose one book uniformly from this complete table. Each item appears once.

Borrowed On shelf
Hardback          8       10
Paperback        15        7

What is P(hardback | borrowed)?

Accepted Answer

8/23. The borrowed group has 8 + 15 = 23 books; 8 are hardbacks. The 18-book hardback row is not the given group.

Question 21

A batch has 60 small and 40 large boxes, with no other sizes. Exactly 1/5 of small boxes and 3/10 of large boxes are marked; the rest are unmarked. Choose one box uniformly. What is P(small | marked)?

Accepted Answer

1/2. There are (1/5) × 60 = 12 marked small boxes and (3/10) × 40 = 12 marked large boxes. The marked group has 24 boxes, so use 12/24.

Question 22

Choose one label uniformly from this complete table. Each item appears once.

Stamped Unstamped
Purple            0         0
Not purple        6        10

What is P(stamped | purple) under the count-ratio rule?

Accepted Answer

Undefined. The purple group is empty, so the ratio would be 0/0. This rule requires a positive conditioning-group count.

Question 23

Choose one tile uniformly from this complete table. Each item appears once.

Marked Unmarked
Round           7        3
Not round       5        9

What is P(not round | unmarked)?

Accepted Answer

3/4. The unmarked group has 3 + 9 = 12 tiles; 9 are not round. Use 9/12 = 3/4.

Question 24

Choose one parcel uniformly from this complete table. Each item appears once.

Sealed Open
Small       6    9
Large      10    5

A learner writes P(small) = 6/16. What fraction gives P(small), with no condition?

Accepted Answer

1/2. All 15 small parcels count, out of 30 total: 15/30 = 1/2. The 6/16 ratio is P(small | sealed).

Question 25

Choose one order uniformly from this complete table. Each item appears once.

Printed Unprinted
Early           11         4
Not early        7         8

What is P(printed | not early)?

Accepted Answer

7/15. The not-early row has 7 + 8 = 15 orders; 7 are printed.

Question 26

A complete collection has 80 cards. Exactly 3/8 of all cards are both blue and dotted. How many cards are both blue and dotted?

Accepted Answer

30 cards. The rate uses the whole collection: (3/8) × 80 = 30.

Question 27

For uniform selection from a complete pin collection, P(round | gold) = 1/4. A learner labels 3/4 as P(round | not gold). Which conditional probability does 3/4 actually give?

Accepted Answer

P(not round | gold). Take the complement of the requested event while keeping the condition gold: 1 − 1/4 = 3/4. The non-gold group can have a different rate.

Question 28

In a nonempty red-tag group, exactly 2/3 are square. No other counts or rates are supplied. Can you determine the fraction of square tags that are red?

Accepted Answer

No. You need the count that is both red and square and the total square count, or equivalent proportions. The within-red rate alone does not determine the reverse condition.

Question 29

Choose one notebook uniformly from this complete table. Each item appears once.

Lined Blank
Small       2     6
Medium      5     3
Large       3     1

What is P(medium | lined)?

Accepted Answer

1/2. The lined column contains 2 + 5 + 3 = 10 notebooks; 5 are medium. Use 5/10.

Question 30

Uniform selection from a complete list of 50 workshop attendees gives P(brings pen | morning session) = 4/5. To which population does this result directly apply?

Accepted Answer

The morning-session attendees in that 50-person list. This fraction alone does not establish the rate for attendees at other workshops.

Question 31

Choose one tile uniformly from this complete table. Each item appears once.

Dot No dot
Blue        5      7
Not blue    4      8

What is P(dot), without a given color?

Accepted Answer

3/8. There are 5 + 4 = 9 dotted tiles among 24 total. Use 9/24 = 3/8.

Question 32

For uniform selection from a collection of tags, P(square | red) = 2/5. A learner says, “Two fifths of the square tags are red.” What group should replace “square tags”?

Accepted Answer

The red tags. The result says two fifths of red tags are square. It does not give P(red | square).

Question 33

Choose one ribbon uniformly from this complete table. Each item appears once.

Dot No dot
Red       4      6
Blue      2      3
Yellow    6      9

A learner uses 4/(4 + 2) for P(red | dot). What fraction corrects the missing category?

Accepted Answer

1/3. Include every dotted ribbon: 4 + 2 + 6 = 12. Use 4/12 = 1/3.

Question 34

A complete tray has 10 red and 10 non-red pieces. The selector is more likely to choose each red piece than each non-red piece. Can counts alone justify P(red) = 10/20?

Accepted Answer

No. The count ratio needs uniform item selection. With unequal selection probabilities, use the pieces’ selection weights; equal counts do not imply equal chances.

Question 35

Choose one basket uniformly from this complete table. Each item appears once.

Lid No lid
Round        4     10
Not round    9      5

What is P(no lid | round)?

Accepted Answer

5/7. The round row contains 4 + 10 = 14 baskets; 10 have no lid. Use 10/14 = 5/7.

Question 36

Under uniform selection from a complete collection of items, P(sealed | blue) = 3/5 and P(blue | sealed) = 1/2. Which group does the 1/2 rate describe?

Accepted Answer

The sealed items. Half of that group is blue; the 3/5 rate instead describes the blue group.

Question 37

A complete list contains only morning (20) and afternoon (50) entries. Exactly 10 in each group are checked; the rest are unchecked. A learner says the checked fractions are equal because both counts are 10. Which group has the larger checked fraction?

Accepted Answer

The morning group. Its fraction is 10/20 = 1/2, while the afternoon fraction is 10/50 = 1/5. Compare within-group rates, not raw counts.

Question 38

A complete collection has 25 red and 15 non-red tokens. Exactly 18 tokens have dots, including 8 red tokens. Every other token has no dot. How many non-red tokens have dots?

Accepted Answer

10 tokens. The dotted column splits into red and non-red: 18 − 8 = 10.

Question 39

Choose one entry uniformly from this complete table. Each item appears once.

Checked Unchecked
Bus          6         4
Walk         3         7
Cycle        9         1

What is P(not bus | checked)?

Accepted Answer

2/3. The checked group contains 6 + 3 + 9 = 18 entries. The walk and cycle entries give 3 + 9 = 12 non-bus entries, so use 12/18 = 2/3.

Question 40

In an observational club list, 4/5 of members who attend a workshop finish a project, versus 2/5 of those who do not attend. Does this comparison alone prove that workshop attendance causes project completion?

Accepted Answer

No. The conditional rates describe an association in that list. Other differences between the groups can explain it; the comparison alone does not establish causation.

Question 41

Choose one bookmark uniformly from this complete table. Each item appears once.

Folded Flat
Orange           7    5
Not orange       3    9

What is P(orange and folded)?

Accepted Answer

7/24. The joint event contains 7 bookmarks out of all 24. Neither a row nor a column is specified as a given condition.

Question 42

Choose one card uniformly from this complete table. Each item appears once.

Star No star
Blue         8       6
Not blue     5      11

What count belongs in the denominator of P(blue | star)?

Accepted Answer

13. The given group is the star column: 8 + 5 = 13 cards.

Question 43

A complete group has 300 items. Its marked percentage is reported as 33%, rounded to the nearest whole percent. Must exactly 99 items be marked?

Accepted Answer

No. A rounded rate does not fix an exact count. Both 98/300 ≈ 32.67% and 99/300 = 33% round to 33%.

Question 44

Choose one chip uniformly from this complete table. Each item appears once.

Marked Unmarked
Red           3        5
Not red       4        8

A learner writes P(marked | red) = 8/3. What fraction corrects the inverted ratio?

Accepted Answer

3/8. There are 3 marked red chips out of 3 + 5 = 8 red chips. A conditional probability cannot exceed 1.

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Conditional Probability Flashcards: Two-Way Table Practice