EST I HS Math · Unit 2 · Data Analysis & Probability
Lesson 2.7 · Probability
On the paper: One of the richest real-question topics — usually 3-5 per paper
Big idea. Every probability question is (favorable outcomes) ÷ (total outcomes) — the whole skill is counting BOTH correctly. Without replacement, the pool shrinks after each draw; for 'different,' compute the complement first and check if it's fewer terms.
1Counting from a bag — then read the question to the end 🎱
🍼 Baby steps
- 1
3 green, 5 red, 8 blue, 4 yellow → 20 total
October 2021. Add every group for the total, even the ones not asked about.
- 2
P(yellow or red) = (4+5) ÷ 20 = 9/20 = 0.45
"Or" between two DIFFERENT colors on one draw just adds the two counts.
- 3
the question asks for the SQUARE of that probability: 0.45² = 0.2025
Read to the very end — "the square of the probability" is a different number from the probability itself.
Trap · the paper's own trap
2Without replacement: the pool shrinks every draw 📦
🍼 Baby steps
- 1
7 balls: 3 red, 2 green, 2 blue. Draw 3, one at a time, NOT replaced.
December 2021. "Successively... without replacing" is the signal.
- 2
ways to get exactly one of each color: 3 × 2 × 2 = 12
One choice from each color group — multiply the group sizes.
- 3
total ways to draw any 3 from 7: (7×6×5) ÷ (3×2×1) = 35
This is "7 choose 3" — divide out the orderings since the draw order inside the 3 doesn't matter here.
- 4
P = 12 ÷ 35
Favorable combinations ÷ total combinations.
Trap · with vs. without replacement changes every denominator
3'Or' from a table, and the complement trick 🔀
🍼 Baby steps
- 1
P(10th grade OR male) = (10th total + male total − both) ÷ grand total
Ahmed Hassan mock. Subtract the overlap ONLY if a student can be counted in both groups — here, "10th grade male" students are.
- 2
P(different colors on 2 spins) = 1 − P(same color on both spins)
Computing "different" directly means adding every mismatched color pair — computing "same" first and subtracting from 1 is almost always fewer terms.
🎛 Play with it · spin it twice
Slide each color's section size and watch P(same) and P(different) update — notice how much shorter the 'same' calculation is, which is exactly why the complement trick saves time.
Spun TWICE — same color: 0.2² + 0.3² + 0.5² = 0.38
Different color: 1 − 0.38 = 0.62
Computing "different" directly means adding up every cross-pair (blue-then-red, red-then-blue, blue-then-yellow…) — far more terms than just computing "same" and subtracting from 1.
4Real EST questions, solved twice 📄
A bag contains 3 green balls, 5 red balls, 8 blue balls, and 4 yellow balls. What is the value of the square of the probability of choosing a yellow or a red ball from the bag?
- A.0.2025
- B.0.45
This is the probability itself — the question asks for its SQUARE.
- C.0.671
No route to this number from the given counts.
- D.0.9
Roughly double 0.45 — not a square.
Safe way · ~25 s
- Total = 3+5+8+4 = 20. P(yellow or red) = 9/20 = 0.45.
- 0.45² = 0.2025.
⏱ 90-second way · ~10 s
- (4+5)/20 = 0.45, then square it: 0.2025.
Answer: A
A box contains 7 identical balls: three red, two green, and two blue. Three balls are drawn randomly and successively, one after the other, WITHOUT replacing the ball in the box. What is the probability of getting exactly one ball of each color?
- A.2/35
Too small — missing a factor from the counting.
- B.12/343
Uses 7³ = 343 in the denominator — that's the WITH-replacement total, not without.
- C.72/343
Same wrong (with-replacement) denominator as above.
- D.12/35
Safe way · ~40 s
- Favorable: 3 × 2 × 2 = 12 (one from each color).
- Total: C(7,3) = (7×6×5)/(3×2×1) = 35.
- P = 12/35.
⏱ 90-second way · ~15 s
- 12 ways to pick one of each color, out of 35 total ways to pick any 3 — 12/35.
Answer: D
A survey of 350 students' favorite subject by education stage shows Secondary totals: Math 80, English 55, Science 10, History 5 (Secondary total 150). If a secondary student is chosen at random, what is the probability that they favor Math or Science?
- A.9/35
Uses the wrong total for the group.
- B.18/25
Too large — bigger than 1 half of the group is unlikely from these counts.
- C.8/15
Close, but not the exact fraction these counts reduce to.
- D.3/5
Safe way · ~20 s
- Math (Secondary) + Science (Secondary) = 80 + 10 = 90.
- 90 ÷ 150 = 3/5.
⏱ 90-second way · ~10 s
- (80+10)/150 simplifies directly to 3/5.
Answer: D
A table of students by grade and gender: 10th grade — 17 female, 18 male; 11th grade — 18 female, 20 male. If a student is selected at random, what is the probability that they are in 10th grade OR male?
- A.0.753
- B.0.521
Forgets to subtract the overlap (10th-grade males), double-counting them.
- C.0.479
This is P(NOT in 10th grade and NOT male) — the wrong event entirely.
- D.0.472
No clean route to this number from the table.
Safe way · ~35 s
- Total = 17+18+18+20 = 73. 10th total = 35. Male total = 38. Both (10th male) = 18.
- P = (35+38−18) ÷ 73 = 55/73 ≈ 0.753.
⏱ 90-second way · ~15 s
- Add the row total and the column total, subtract their overlap once, divide by the grand total.
Answer: A
A spinner has 2 blue sections, 3 red sections, and 5 yellow sections. It is spun twice. What is the probability of getting two different colors?
- A.1/2
A rough guess, not the computed value.
- B.5/3
Bigger than 1 — not a valid probability at all.
- C.31/50
- D.32/50
An arithmetic slip in the same computation.
Safe way · ~35 s
- P(same color) = (2/10)² + (3/10)² + (5/10)² = 0.04+0.09+0.25 = 0.38.
- P(different) = 1 − 0.38 = 0.62 = 31/50.
⏱ 90-second way · ~15 s on the calculator
- Square each color's own probability, add them, subtract from 1 — far fewer terms than listing every mismatched pair.
Answer: C
⏱ The 90-second lane
One question = 90 seconds. These are the moves that finish this topic in one line.
- 1Every probability = (favorable outcomes) ÷ (total outcomes) — count BOTH carefully, and read to the end of the question (it may ask for a square, a sum, or a specific combination of results).
- 2"Or" between two groups on one draw: add the counts. If a person/item can belong to BOTH groups, subtract the overlap once.
- 3Without replacement, the total shrinks by 1 after every draw, and it's a DIFFERENT shrinking total for each draw.
- 4"Exactly k out of n" with equal chances is a counting problem: count how many ways, divide by the total ways.
- 5For "different" outcomes across repeated independent trials, try the complement first: 1 − P(same) — check whether it has fewer terms before computing "different" directly.
Take with you
- ▸Probability is always a counting problem in disguise — get the favorable count and the total count right, and the fraction follows.
- ▸Replacement changes everything: with it, every draw is independent and the pool never shrinks; without it, it does.
- ▸The complement (1 − P(the opposite)) is a genuine shortcut, not just a formula — check if it has fewer terms before committing to a direct computation.