Probability Tree Diagrams

01

BasicWith Replacement

A bag has 3 red and 7 blue marbles. One marble is drawn and replaced, then a second is drawn.
What is the probability of drawing two red marbles?

📌 Quick Memory WITH replacement → fractions stay the same both draws. Just multiply!

P(Red, Red) = 310 \times 310 = ?

02

BasicWith Replacement

A spinner has 4 equal sections: 1 Green, 3 Yellow. You spin it twice (independent).
What is the probability of spinning Yellow both times?

📌 Quick Memory Independent events → WITH replacement logic. Multiply the same probability twice.

P(Y, Y) = 34 \times 34 = ?

03

With ReplacementTRAP ⚠

A bag has 4 red and 5 blue marbles. You draw twice with replacement.
What is the probability of drawing at least one blue?

📌 Quick Memory "At least one" → use the COMPLEMENT trick!
P(at least one blue) = 1 − P(no blue) = 1 − P(red, red)

P(at least 1 blue) = 1 - 49 \times 49 = 1 - 1681 = ?

04

With ReplacementSame Color

A bag has 4 red and 5 blue marbles. Draw twice with replacement.
What is the probability of getting the same color both times?

📌 Quick Memory Same color = (RR) OR (BB) → Add both paths after multiplying each!

P(same) = P(R,R) + P(B,B) = 49 \times 49 + 59 \times 59 = 1681 + 2581 = ?

05

Without ReplacementBasic

A bag has 4 red and 5 blue marbles. Draw twice without replacement.
What is P(Blue, Blue)?

📌 Quick Memory WITHOUT replacement → after 1st draw, TOTAL goes down by 1, and so does the color count if you drew that color!

P(B, B) = 59 \times 48 = ?

06

Without ReplacementTRAP ⚠

A deck has 5 hearts and 3 spades. Two cards are drawn without replacement.
What is the probability the first is a heart and the second is a spade?

📌 Quick Memory Different colors → numerator of 2nd fraction stays the same (spades didn't change), but the TOTAL drops from 8 to 7!

P(H then S) = 58 \times 37 = ?

07

Without ReplacementSame Color

A bag has 3 green and 4 orange. Two marbles drawn without replacement.
What is the probability of getting same color both times?

📌 Quick Memory Same color (no replace) → P(G,G) + P(O,O). Second fraction: color count AND total BOTH decrease by 1!

P(G,G) = 37 \times 26 = 642 P(O,O) = 47 \times 36 = 1242 Total = ?

08

IndependentReal-world

The probability Anna passes her driving test is 0.7. The probability Rob passes is 0.45. They take their tests on the same day (independently).
What is the probability both pass?

📌 Quick Memory Independent events → just MULTIPLY. No adjusting needed because one result doesn't affect the other!

P(Anna passes AND Rob passes) = 0.7 \times 0.45 = ?

09

IndependentTRAP ⚠

Anna passes with P = 0.7, Rob passes with P = 0.45.
What is the probability that Anna passes but Rob fails?

📌 Quick Memory "Fails" = 1 − P(passes). Calculate the complement first, then multiply!

P(Rob fails) = 1 - 0.45 = 0.55 P(Anna passes, Rob fails) = 0.7 \times 0.55 = ?

10

ConditionalHARD ★

Kim plays 2 pickleball games. P(wins game 1) = 0.60.
If he wins game 1 → P(wins game 2) = 0.68.
If he loses game 1 → P(wins game 2) = 0.40.

What is the probability he ends 1 win and 1 loss (a 1-1 record)?

📌 Quick Memory CONDITIONAL probability → the 2nd branch probability CHANGES depending on what happened in the 1st branch. Draw the tree! Two different paths give 1W-1L: Win-Lose OR Lose-Win.

Path 1: Win \to Lose = 0.60 \times (1-0.68) = 0.60 \times 0.32 Path 2: Lose \to Win = (1-0.60) \times 0.40 = 0.40 \times 0.40 P(1-1) = ? + ? = ?

11

WeightedHARD ★

In a grocery store, 20% of items are dairy and 65% are snack food. 23% of dairy items are expired. 10% of snack items are expired.

What is the probability that a randomly chosen item is expired?

📌 Quick Memory "Weighted tree" → multiply each category's share × its expire rate, then ADD them all together. Think: GROUP SIZE × RATE WITHIN GROUP.

P(expired) = P(dairy) \times P(exp|dairy) + P(snack) \times P(exp|snack) = 0.20 \times 0.23 + 0.65 \times 0.10 = 0.046 + 0.065 = ?

12

WeightedReal-world

A factory has two machines. Machine A produces 60% of items; 4% are defective. Machine B produces the rest; 7% are defective.
What is the probability a randomly chosen item is defective?

📌 Quick Memory Machine B produces 40% (= 100% − 60%). Apply the same weighted formula: sum of (share × defect rate).

P(defective) = 0.60 \times 0.04 + 0.40 \times 0.07 = 0.024 + 0.028 = ?

13

TRAP ⚠Replace vs Not

A bag has 6 marbles: 2 red, 4 blue. Two are drawn.
Which scenario gives a higher probability of getting 2 blue marbles — with replacement or without replacement?

📌 Quick Memory With replacement: same fraction both times. Without replacement: 2nd fraction's numerator AND total both drop → usually gives a different (often smaller) result. Calculate and compare!

With replacement: 46 \times 46 = 1636 \approx 0.444 Without replacement: 46 \times 35 = 1230 = 0.400

14

TRAP ⚠Add vs Multiply

A coin is flipped twice. What is the probability of getting exactly one Head?

📌 Quick Memory "Exactly one" → list ALL paths that give 1 Head: HT and TH. Multiply along each path, then ADD the two paths.

P(HT) = 12 \times 12 = 14 P(TH) = 12 \times 12 = 14 Total = ?

15

ComplementHARD ★

A bag has 2 red, 3 blue, 5 green marbles (10 total). Two are drawn with replacement.
What is the probability of getting at least one green?

📌 Quick Memory "At least one green" is HARD to count directly. Use complement: P(at least 1 green) = 1 − P(NO green). P(no green) = P(not green, not green).

P(not green) = 510 = 12 P(no green twice) = 12 \times 12 = 14 P(at least 1 green) = 1 - 14 = ?

16

Without ReplacementHARD ★

A class has 3 boys and 5 girls. Two students are selected without replacement.
What is the probability the second student is a girl given the first was a girl?

📌 Quick Memory "GIVEN that" = conditional probability. After one girl is picked, 4 girls remain out of 7 total. The condition already happened — just look at what's left!

P(2nd girl | 1st girl) = 47

17

Three EventsHARD ★

A fair coin is flipped 3 times. What is the probability of getting exactly 2 Heads?

📌 Quick Memory List ALL arrangements of 2H 1T: HHT, HTH, THH. Each has probability (½)³ = ⅛. Count the arrangements × single path probability.

Arrangements with 2 heads: HHT, HTH, THH \to 3 ways Each probability: 12 \times 12 \times 12 = 18 Total = 3 \times 18 = ?

18

Probability Sum CheckTRAP ⚠

A student says: "I drew a tree diagram and found P(RR) = 1/4, P(RB) = 1/4, P(BR) = 1/4, P(BB) = 1/3."
What is wrong with this?

📌 Quick Memory ALL branches of a complete tree diagram MUST add up to exactly 1. If they don't, something is wrong — either a calculation error or a missing branch!

14 + 14 + 14 + 13 = 312 + 312 + 312 + 412 = 1312 \neq 1

19

Multi-stepHARD ★

Box A has 4 red, 2 blue. Box B has 1 red, 5 blue. A box is chosen at random (50/50), then one marble is drawn.
What is the probability of drawing a red marble?

📌 Quick Memory Two-stage tree: first pick the BOX, then pick the COLOR. P(red) = P(Box A)×P(red|A) + P(Box B)×P(red|B). This is the "weighted tree" again!

P(red) = 12 \times 46 + 12 \times 16 = 412 + 112 = ?

20

Final BossHARD ★★

A bag has 5 red and 3 blue marbles. Three marbles are drawn without replacement.
What is the probability that all three are the same color?

📌 Quick Memory Three draws, no replace → multiply THREE fractions. Same color = P(RRR) + P(BBB). For blue: only 3 total so after 2 blues there's only 1 left!

P(RRR) = 58 \times 47 \times 36 = 60336 P(BBB) = 38 \times 27 \times 16 = 6336 Total = ?