Tree Diagrams (B) Worksheet | PDF Printable Statistics ... - Free Printable

Let’s solve each problem step by step.

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Problem 1: Cubes in a box (without replacement)

We have:
- 12 red cubes (R)
- 6 green cubes (G)
→ Total = 18 cubes

First choice:
- P(R) = 12/18 = 2/3
- P(G) = 6/18 = 1/3

Second choice (depends on first):

If first was R → now 11 R and 6 G left → total 17
- P(R|R) = 11/17
- P(G|R) = 6/17

If first was G → now 12 R and 5 G left → total 17
- P(R|G) = 12/17
- P(G|G) = 5/17

Now calculate:

a. Two red cubes

That means: First R AND Second R
= P(R) × P(R|R) = (12/18) × (11/17)
= (2/3) × (11/17) = 22/51

✔ Check: 12×11 = 132, 18×17 = 306 → simplify 132/306 ÷ 6 = 22/51 ✔️

b. At least one green cube

This is easier to do as:
1 – P(no green) = 1 – P(two reds)
= 1 – 22/51 = 29/51

✔ Alternatively, add up all paths with at least one G:
- R then G: (12/18)(6/17) = 72/306
- G then R: (6/18)(12/17) = 72/306
- G then G: (6/18)(5/17) = 30/306
Total = (72 + 72 + 30)/306 = 174/306 = 29/51 ✔️

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Problem 2: Boys and girls in class

20 boys (B), 10 girls (G) → total 30 students

Two chosen at random, without replacement.

Tree diagram probabilities:

First choice:
- P(B) = 20/30 = 2/3
- P(G) = 10/30 = 1/3

Second choice:

If first was B → 19 B, 10 G left → total 29
- P(B|B) = 19/29
- P(G|B) = 10/29

If first was G → 20 B, 9 G left → total 29
- P(B|G) = 20/29
- P(G|G) = 9/29

a. If first student is a girl, what’s probability second is also a girl?

This is conditional: given first is G, find P(second is G)

From above: P(G|G) = 9/29

✔ Answer: 9/29

b. Probability that a girl and boy are chosen?

This can happen two ways:
- Girl then Boy: P(G) × P(B|G) = (10/30) × (20/29) = (1/3)(20/29) = 20/87
- Boy then Girl: P(B) × P(G|B) = (20/30) × (10/29) = (2/3)(10/29) = 20/87

Add them: 20/87 + 20/87 = 40/87

✔ Check: 10×20 = 200, 30×29 = 870 → 200/870 = 20/87 per path → total 40/87 ✔️

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Problem 3: Nails in a box

Total nails: 200
Too small (S): 5
Not too small (S’): 195

Three nails selected at random, without replacement.

Tree has three levels: First nail, Second nail, Third nail.

We’ll need conditional probabilities for each branch.

But let’s answer the questions directly.

a. If first nail is too small, what’s probability second is also too small?

Given first is S → now 4 S left, 199 total nails

So P(S | first was S) = 4/199

✔ Answer: 4/199

b. All three nails are too small

P(all three S) = P(first S) × P(second S | first S) × P(third S | first two S)

= (5/200) × (4/199) × (3/198)

Calculate step by step:

5/200 = 1/40
4/199 stays
3/198 = 1/66

So: (1/40) × (4/199) × (1/66) = (1 × 4 × 1) / (40 × 199 × 66)

Numerator: 4
Denominator: 40 × 199 × 66

First, 40 × 66 = 2640
Then 2640 × 199

Let’s compute 2640 × 200 = 528,000 → subtract 2640 → 528,000 - 2,640 = 525,360

So denominator = 525,360

Fraction: 4 / 525,360 = simplify by dividing numerator and denominator by 4:

= 1 / 131,340

✔ Final answer: 1/131340

c. Only the second nail is too small

That means: First NOT small, Second IS small, Third NOT small

Path: S’ → S → S’

Probabilities:

P(first S’) = 195/200
P(second S | first S’) = 5/199 (since one non-small removed, still 5 small left, total 199)
P(third S’ | first S’, second S) = 194/198 (because we’ve removed one S’ and one S, so 194 S’ left out of 198 total)

So multiply:

(195/200) × (5/199) × (194/198)

Let’s simplify step by step.

First, 195/200 = 39/40 (divide by 5)
5/199 stays
194/198 = 97/99 (divide by 2)

So: (39/40) × (5/199) × (97/99)

Multiply numerators: 39 × 5 × 97
Multiply denominators: 40 × 199 × 99

Compute numerator:

39 × 5 = 195
195 × 97 → let’s do 200×97 = 19,400 minus 5×97=485 → 19,400 - 485 = 18,915

Denominator:

40 × 199 = 7,960
7,960 × 99 = 7,960 × 100 - 7,960 = 796,000 - 7,960 = 788,040

So fraction: 18,915 / 788,040

Can we simplify?

Check if both divisible by 15? Let’s see:

18,915 ÷ 5 = 3,783 → yes
788,040 ÷ 5 = 157,608 → yes

Now 3,783 / 157,608

Check divisibility by 3:

3+7+8+3=21 → divisible by 3
1+5+7+6+0+8=27 → divisible by 3

Divide numerator and denominator by 3:

3,783 ÷ 3 = 1,261
157,608 ÷ 3 = 52,536

Now check 1,261 and 52,536 — any common factors?

Try dividing 1,261 by small primes:

1,261 ÷ 13 = 97 → because 13×97 = 1,261? Let’s check: 13×90=1,170; 13×7=91 → 1,170+91=1,261 ✔️

So 1,261 = 13 × 97

Now check if 52,536 divisible by 13 or 97?

Try 52,536 ÷ 13:

13 × 4,000 = 52,000 → remainder 536
13 × 41 = 533 → remainder 3 → not divisible

Try 97: 97 × 541 = ? Maybe too big — perhaps no more simplification.

Wait — maybe I made it too complicated. Let’s go back.

Original: (195/200) × (5/199) × (194/198)

Note: 195 = 5×39, 194 = 2×97, etc.

But perhaps leave as simplified fraction after canceling common factors early.

Alternative approach:

Write as:

(195 × 5 × 194) / (200 × 199 × 198)

Factor:

195 = 5 × 39 = 5 × 3 × 13
5 = 5
194 = 2 × 97

Numerator: 5 × 3 × 13 × 5 × 2 × 97 = 2 × 3 × 5² × 13 × 97

Denominator:

200 = 2³ × 5²
199 = prime
198 = 2 × 99 = 2 × 9 × 11 = 2 × 3² × 11

So denominator: 2³ × 5² × 199 × 2 × 3² × 11 = 2⁴ × 3² × 5² × 11 × 199

Now cancel common factors:

Numerator: 2¹ × 3¹ × 5² × 13 × 97
Denominator: 2⁴ × 3² × 5² × 11 × 199

Cancel 2¹, 3¹, 5²

Left:

Numerator: 13 × 97
Denominator: 2³ × 3¹ × 11 × 199 = 8 × 3 × 11 × 199

Compute denominator: 8×3=24; 24×11=264; 264×199

264×200 = 52,800 → minus 264 = 52,536

Numerator: 13×97 = 1,261

So same as before: 1,261 / 52,536

And since 1,261 = 13×97, and 52,536 not divisible by 13 or 97 (as checked earlier), this is simplified.

✔ So final answer: 1261/52536

But wait — let me double-check calculation of denominator:

Earlier I had 200 × 199 × 198 = ?

200 × 198 = 39,600
39,600 × 199 = 39,600 × 200 - 39,600 = 7,920,000 - 39,600 = 7,880,400? Wait — that doesn’t match previous.

I think I messed up earlier.

Let me recalculate denominator properly:

200 × 199 × 198

First, 200 × 198 = 39,600
Then 39,600 × 199

Better: 39,600 × 200 = 7,920,000
Minus 39,600 = 7,920,000 - 39,600 = 7,880,400

Numerator: 195 × 5 × 194

195 × 5 = 975
975 × 194

Compute 975 × 200 = 195,000
Minus 975 × 6 = 5,850 → 195,000 - 5,850 = 189,150

So fraction: 189,150 / 7,880,400

Now simplify.

Divide numerator and denominator by 30? Or find GCD.

First, divide by 10: 18,915 / 788,040 — same as before.

Then as before, divided by 15? Earlier we got to 1,261 / 52,536

But 189,150 ÷ 150 = ? Perhaps better to use calculator-style reduction.

Notice:

189,150 / 7,880,400 = ?

Divide numerator and denominator by 150:

189,150 ÷ 150 = 1,261
7,880,400 ÷ 150 = 52,536

Yes! Because 150 × 1,261 = 189,150? 150×1,200=180,000; 150×61=9,150 → total 189,150 ✔️
150 × 52,536 = 7,880,400? 150×50,000=7,500,000; 150×2,536=380,400 → total 7,880,400 ✔️

So yes, 1,261 / 52,536

And as established, no further simplification.

✔ So answer is 1261/52536

But let me confirm if this reduces — try dividing 1261 and 52536 by... say 13? 1261÷13=97, as before. 52536÷13: 13×4041=52,533 → remainder 3 → not divisible. By 97? 97×541=52,477; 52,536 - 52,477=59 → not divisible. So yes, simplified.

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Final Answers:

Problem 1:
a. 22/51
b. 29/51

Problem 2:
a. 9/29
b. 40/87

Problem 3:
a. 4/199
b. 1/131340
c. 1261/52536

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Final Answer:
1a. \frac{22}{51}
1b. \frac{29}{51}
2a. \frac{9}{29}
2b. \frac{40}{87}
3a. \frac{4}{199}
3b. \frac{1}{131340}
3c. \frac{1261}{52536}