SAT-59: Conditional Probability from Two-Way Tables

SAT-59: Conditional Probability from Two-Way Tables

Description: A two-way table sorts data by two categories at once (for example, gender × choice). Conditional probability asks for the chance of something given that we already know one category. The whole skill is choosing the right denominator.

Reading a two-way table

Rows are one category, columns are the other, and each cell is a count. The margins (the "Total" row and column) give the totals for each category and the grand total. (Oʻzbekcha: ikki tomonlama jadval maʼlumotni bir vaqtda ikki belgi boʻyicha ajratadi.)

The key idea — "given" changes the denominator

Ordinary probability uses the grand total as the denominator. Conditional probability ("given that…") restricts you to just one row or one column, so that row/column total becomes the new denominator. (Oʻzbekcha: "shart bilan" (given) — butun jamiga emas, faqat bitta qator yoki ustun jamiga boʻlamiz.)

Our example table

A survey of 100 students asked if they prefer Tea or Coffee:

• Boys: Tea 20, Coffee 30 (row total 50).
• Girls: Tea 35, Coffee 15 (row total 50).
• Column totals: Tea 55, Coffee 45; grand total 100.

Worked Example 1 — ordinary probability

What is the probability a random student prefers Tea?

• Use the grand total: P(Tea) = 55/100 = 0.55.

Worked Example 2 — conditional (given a row)

Given that a student is a Boy, what is the probability he prefers Coffee?

• "Given Boy" → restrict to the Boys row (total 50), not 100.
• Boys who chose Coffee = 30. So P(Coffee | Boy) = 30/50 = 0.6.

(Oʻzbekcha: "oʻgʻil bola sharti bilan" — faqat oʻgʻil bolalar qatoriga (50) boʻlamiz.)

Worked Example 3 — conditional (given a column)

Given that a student prefers Tea, what is the probability the student is a Girl?

• "Given Tea" → restrict to the Tea column (total 55).
• Girls in the Tea column = 35. So P(Girl | Tea) = 35/55 = 7/11 ≈ 0.64.

The trap: P(Coffee | Boy) and P(Boy | Coffee) are different questions with different denominators. Read which group is "given." (Oʻzbekcha: shartni diqqat bilan oʻqing — maxraj oʻzgaradi.)

A reliable two-step routine

Conditional-probability questions become almost automatic if you always do two things in order:

1. Find the denominator first. The word after "given" names a single row or column — write down that total before anything else. This is the most common place students slip.
2. Find the numerator inside that group. Count only the members of the given row/column that also satisfy the second condition.

Because you have already narrowed to one row or column, you never use the grand total in a conditional question. If you ever find yourself dividing by the grand total, you have probably ignored the word "given." (Oʻzbekcha: avval maxrajni (berilgan qator yoki ustun jamisini) yozing, keyin shu guruh ichidagi kerakli sonni toping.)

Why this differs from "and" probability

"P(Boy and Coffee)" divides by the grand total (100), giving 30/100 = 0.3. "P(Coffee given Boy)" divides by the Boys total (50), giving 30/50 = 0.6. Same cell, different denominators — so always separate "and" from "given." (Oʻzbekcha: "va" — umumiy jamiga, "given" — faqat bitta guruh jamiga boʻlinadi.)

Practice 1

Given that a student is a Girl, what is the probability she prefers Tea?

Practice 2

Given that a student prefers Coffee, what is the probability the student is a Boy?

Key words — Kalit soʻzlar

• Two-way table — ikki tomonlama jadval
• Conditional probability — shartli ehtimollik
• Given (that) — shart bilan (berilganda)
• Row / Column — qator / ustun
• Cell — katak
• Margin total — chekka (yon) jami
• Grand total — umumiy jami
• Frequency — chastota
• Denominator — maxraj

Summary

• In a two-way table, cells are counts; margins are category totals.
• Ordinary probability divides by the grand total.
• Conditional ("given X") divides by the total of just that row or column.
• P(A | B) and P(B | A) are different — check which group is "given."