Statistics & Probability
Two-Way Tables Probability
Two-way tables organize data by two categories. The KEY skill is knowing what goes on the bottom of your fraction — and that depends on the type of question: • P(event) — no "given" — bottom is the GRAND TOTAL (whole survey). • P(A AND B) — both things happen — bottom is the GRAND TOTAL. • P(A | B) — "given B" — bottom is the TOTAL FOR B ONLY. You zoom in to just that row or column and ignore everyone else. The word GIVEN tells you to shrink your world. Instead of the whole survey, you only look at the group that was given to you.
Watch & Learn
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Worked Examples
Follow along with these step-by-step examples. Take your time — there's no rush!
Problem
A survey of 100 students asked about gender and whether they like math. What is P(likes math)? No "given" here — we are asking about ALL 100 students. Bottom = grand total.
Likes Math?
| Gender | Yes | No | Total |
|---|---|---|---|
| Boy | 25 | 15 | 40 |
| Girl | 35 | 25 | 60 |
| Total | 60 | 40 | 100 |
✅ Bottom = 100 (grand total — no GIVEN)
Step-by-Step Solution
No "given" in this question → use the GRAND TOTAL (100) on the bottom.
Students who like math = Yes column total (green): 60.
P(likes math) = 60 ÷ 100 = 0.6
Answer
P(likes math) = 60/100 = 0.6
Problem
What is P(boy AND likes math)? AND means both things are true at once. Find the one cell where they overlap. Bottom = grand total.
Likes Math?
| Gender | Yes | No | Total |
|---|---|---|---|
| Boy | 25 | 15 | 40 |
| Girl | 35 | 25 | 60 |
| Total | 60 | 40 | 100 |
✅ Bottom = 100 (grand total — AND question, no GIVEN)
Step-by-Step Solution
AND question → find the single overlapping cell: boys who like math = 25 (gold).
Bottom = grand total = 100.
P(boy AND likes math) = 25 ÷ 100 = 0.25
Answer
P = 25/100 = 0.25
Problem
What is P(likes math | girl)? ⚠️ The word GIVEN changes everything. "Given girl" means we ONLY look at girls — we shrink our world to the Girl row. Bottom = total girls, NOT 100.
Likes Math?
| Gender | Yes | No | Total |
|---|---|---|---|
| Boy | 25 | 15 | 40 |
| Girl | 35 | 25 | 60 |
| Total | 60 | 40 | 100 |
⚠️ Bottom = 60 (total girls only — GIVEN girl shrinks your world)
Step-by-Step Solution
"Given girl" → zoom in to the Girl row only (blue). Forget the boys.
Total girls = 60. This is your new bottom.
Girls who like math (gold cell) = 35.
P(likes math | girl) = 35 ÷ 60 ≈ 0.583
❌ Wrong answer if you use 100: 35/100 = 0.35 — that ignores the GIVEN.
Answer
P(likes math | girl) = 35/60 ≈ 0.583
Problem
What is P(boy | does not like math)? ⚠️ GIVEN "does not like math" — shrink your world to the No column only. Bottom = total who said No.
Likes Math?
| Gender | Yes | No | Total |
|---|---|---|---|
| Boy | 25 | 15 | 40 |
| Girl | 35 | 25 | 60 |
| Total | 60 | 40 | 100 |
⚠️ Bottom = 40 (total who said No — GIVEN shrinks your world)
Step-by-Step Solution
"Given does not like math" → zoom in to the No column (green). Forget everyone else.
Total who said No = 40. This is your new bottom.
Boys who said No (gold cell) = 15.
P(boy | does not like math) = 15 ÷ 40 = 0.375
❌ Wrong answer if you use 100: 15/100 = 0.15 — that ignores the GIVEN.
Answer
P(boy | does not like math) = 15/40 = 0.375
Problem
Quick check — what is the difference between P(girl AND likes math) and P(likes math | girl)?
Likes Math?
| Gender | Yes | No | Total |
|---|---|---|---|
| Boy | 25 | 15 | 40 |
| Girl | 35 | 25 | 60 |
| Total | 60 | 40 | 100 |
Same gold cell (35) — different bottom number!
Step-by-Step Solution
Both questions point to the same gold cell: girls who like math = 35.
P(girl AND likes math) = 35 ÷ 100 = 0.35 ← bottom is grand total
P(likes math | girl) = 35 ÷ 60 ≈ 0.583 ← bottom is total girls (GIVEN)
Same top number, different bottom — very different answers!
Answer
P(AND) = 35/100 = 0.35 vs. P(GIVEN) = 35/60 ≈ 0.583
Your Turn — Practice Problems
Try all 5 problems on your own first. Write out your work — that's how it sticks!
💡 Tip: Don't peek at the answers until you've genuinely tried each one.
A shelter surveyed 80 pets. What is P(indoor pet)? No "given" — use the grand total.
Location
| Animal | Indoor | Outdoor | Total |
|---|---|---|---|
| Cat | 20 | 10 | 30 |
| Dog | 15 | 35 | 50 |
| Total | 35 | 45 | 80 |
✅ Bottom = 80 (grand total — no GIVEN)
Same table. What is P(cat AND outdoor)? AND question — use the grand total.
Location
| Animal | Indoor | Outdoor | Total |
|---|---|---|---|
| Cat | 20 | 10 | 30 |
| Dog | 15 | 35 | 50 |
| Total | 35 | 45 | 80 |
✅ Bottom = 80 (grand total — AND question)
Same table. What is P(dog | outdoor)? GIVEN outdoor — shrink to the Outdoor column only.
Location
| Animal | Indoor | Outdoor | Total |
|---|---|---|---|
| Cat | 20 | 10 | 30 |
| Dog | 15 | 35 | 50 |
| Total | 35 | 45 | 80 |
⚠️ Bottom = 45 (total outdoor only — GIVEN shrinks your world)
100 people surveyed about exercise and health. What is P(healthy)? No "given" — grand total.
Healthy?
| Exercise? | Healthy | Not Healthy | Total |
|---|---|---|---|
| Exercises | 30 | 10 | 40 |
| Does Not | 15 | 45 | 60 |
| Total | 45 | 55 | 100 |
✅ Bottom = 100 (grand total — no GIVEN)
Same table. What is P(healthy | exercises)? GIVEN exercises — shrink to the Exercises row only.
Healthy?
| Exercise? | Healthy | Not Healthy | Total |
|---|---|---|---|
| Exercises | 30 | 10 | 40 |
| Does Not | 15 | 45 | 60 |
| Total | 45 | 55 | 100 |
⚠️ Bottom = 40 (total who exercise — GIVEN shrinks your world)
Same table. What is P(exercises AND healthy)? AND question — grand total.
Healthy?
| Exercise? | Healthy | Not Healthy | Total |
|---|---|---|---|
| Exercises | 30 | 10 | 40 |
| Does Not | 15 | 45 | 60 |
| Total | 45 | 55 | 100 |
✅ Bottom = 100 (grand total — AND question)
Same table. What is P(does not exercise | not healthy)? GIVEN not healthy — shrink to Not Healthy column.
Healthy?
| Exercise? | Healthy | Not Healthy | Total |
|---|---|---|---|
| Exercises | 30 | 10 | 40 |
| Does Not | 15 | 45 | 60 |
| Total | 45 | 55 | 100 |
⚠️ Bottom = 55 (total not healthy — GIVEN shrinks your world)
100 students surveyed about sports. What is P(plays sports)? No "given" — grand total.
Plays Sports?
| Gender | Yes | No | Total |
|---|---|---|---|
| Boy | 20 | 30 | 50 |
| Girl | 15 | 35 | 50 |
| Total | 35 | 65 | 100 |
✅ Bottom = 100 (grand total — no GIVEN)
Same table. What is P(girl AND plays sports)? AND question — grand total.
Plays Sports?
| Gender | Yes | No | Total |
|---|---|---|---|
| Boy | 20 | 30 | 50 |
| Girl | 15 | 35 | 50 |
| Total | 35 | 65 | 100 |
✅ Bottom = 100 (grand total — AND question)
Same table. What is P(plays sports | boy)? GIVEN boy — shrink to the Boy row only.
Plays Sports?
| Gender | Yes | No | Total |
|---|---|---|---|
| Boy | 20 | 30 | 50 |
| Girl | 15 | 35 | 50 |
| Total | 35 | 65 | 100 |
⚠️ Bottom = 50 (total boys — GIVEN shrinks your world)
Same table. What is P(boy | does not play sports)? GIVEN does not play — shrink to the No column.
Plays Sports?
| Gender | Yes | No | Total |
|---|---|---|---|
| Boy | 20 | 30 | 50 |
| Girl | 15 | 35 | 50 |
| Total | 35 | 65 | 100 |
⚠️ Bottom = 65 (total who don't play — GIVEN shrinks your world)
What is the rule? When do you use the grand total vs. the category total on the bottom?
What goes on the bottom?
| Question Type | Bottom number | Example |
|---|---|---|
| P(A) — no given | Grand total | 60/100 |
| P(A AND B) | Grand total | 25/100 |
| P(A | B) — GIVEN | Category total only | 35/60 |
Gold row = GIVEN questions — bottom shrinks to the given category
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