Math worksheet focused on compound probability, featuring problems that involve calculating the likelihood of combined events using spinners, marbles, and multiple-choice scenarios.

Let’s go through each problem one by one. I’ll solve them step by step, check my work, and then give you the final answers at the end.

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Problem 1:
> What is the difference between independent events and dependent events?
> With independent events, the first event does not affect the second event.
> With dependent events, the first event does have an effect on the second event.

✔ This is already answered correctly in the worksheet. No calculation needed — it’s a definition question.

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Problem 2:
> A coin is tossed and a die is rolled. What is the probability of tossing tails and rolling either a 5 or 6?

- P(tails) = 1/2
- P(roll 5 or 6) = 2 out of 6 = 1/3
- Since these are independent: multiply probabilities → (1/2) × (1/3) = 1/6

But wait — the student wrote “1/2 x 2/6 = 1/6” — that’s also correct because 2/6 simplifies to 1/3, so same result.

✔ Answer: 1/6

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Problem 3:
> A bag contains 4 red lollipops and 6 blue lollipops. What is the probability of randomly picking two blue lollipops if they are replaced after each pick?

Total lollipops = 4 + 6 = 10

P(first blue) = 6/10
Since replaced, P(second blue) = 6/10 again

So: (6/10) × (6/10) = 36/100 = 9/25

Student wrote: 6/10 × 6/10 = 36/100 → which reduces to 9/25. Correct!

✔ Answer: 9/25

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Problem 4:
> Dupree flips a coin four times. What is the probability that it lands on tails all four times?

Each flip: P(tails) = 1/2
Four independent flips: (1/2)^4 = 1/16

✔ Answer: 1/16

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Problem 5:
> If you spin the spinner to the right twice, what is the probability of spinning 3 both times?

Spinner has sections: Let’s count from image description (even though we can’t see it, student wrote 1/5 x 1/5).

Assuming spinner has 5 equal sections, and only one section is labeled “3”.

So P(spinner shows 3) = 1/5 per spin.

Two spins: (1/5) × (1/5) = 1/25

Student wrote: 1/5 x 1/5 = 1/25 → correct.

✔ Answer: 1/25

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Problem 6:
> An ice cream shop has 2 pints of chocolate, 3 pints of vanilla, and 1 pint of strawberry. Use this to answer questions 6–7.

Total pints = 2 + 3 + 1 = 6

Question 6:
> Carl randomly draws a sample of one for the taste test, and then draws another marble. What is the probability that he will draw chocolate and then vanilla?

Wait — says “draws another marble”? Probably typo — should be “pint”. And since no replacement mentioned, assume without replacement.

P(first chocolate) = 2/6
After removing one chocolate, total left = 5, vanilla still 3 → P(second vanilla) = 3/5

So: (2/6) × (3/5) = 6/30 = 1/5

Student wrote: 2/6 x 3/5 = 6/30 → which is 1/5. Correct.

✔ Answer: 1/5

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Problem 7:
> Same setup as #6. What is the probability that he will draw vanilla and then chocolate?

P(first vanilla) = 3/6
Then P(chocolate) = 2/5 (since one vanilla removed, total 5 left, chocolate still 2)

So: (3/6) × (2/5) = 6/30 = 1/5

Same as above — commutative in this case.

Student didn’t write answer here, but logically it’s same as #6.

✔ Answer: 1/5

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Problem 8:
> Aaron randomly draws a marble out of the bag, puts it back, and then draws another marble. What is the probability that he will draw a green marble and then a black marble?

Bag contents: 13 green, 12 black → total = 25 marbles

With replacement → independent events.

P(green) = 13/25
P(black) = 12/25

Multiply: (13/25) × (12/25) = 156 / 625

Check if reducible: 156 and 625 share no common factors (625=5^4, 156=2²×3×13) → stays 156/625

Student wrote: 13/25 x 12/25 = 156/625 → correct.

✔ Answer: 156/625

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Problem 9:
> Mr. Shepard’s class has 13 girls and 12 boys. He is drawing names to decide the order in which they present their projects. What is the probability that he draws a girl first and then a boy?

Total students = 13 + 12 = 25

No replacement mentioned → assume without replacement

P(first girl) = 13/25
Then P(boy) = 12/24 (since one girl removed, total 24 left, boys still 12)

So: (13/25) × (12/24) = (13/25) × (1/2) = 13/50

Wait — student wrote: 13/25 x 12/24 = 156/600 → let’s simplify:

156 ÷ 12 = 13, 600 ÷ 12 = 50 → yes, 13/50

But student wrote 156/600 — which is correct before simplifying. Either form acceptable unless specified.

We’ll use simplified: 13/50

✔ Answer: 13/50

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Problem 10:
> In a set of 100 batteries, 3 do not work. A product testing person picks two batteries to test. What is the probability that both batteries picked do not work?

Total batteries = 100
Defective = 3

Without replacement.

P(first defective) = 3/100
P(second defective) = 2/99 (one defective already taken)

So: (3/100) × (2/99) = 6 / 9900

Simplify: divide numerator and denominator by 6 → 1 / 1650

Student wrote: 3/100 x 2/99 = 6/9900 → correct, and 6/9900 = 1/1650

✔ Answer: 1/1650

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Problem 11:
> You are taking a quiz with three multiple choice questions. Each question has four answer choices. What is the probability that you answer each question correctly if you randomly guess the answer to each question?

Each question: P(correct) = 1/4
Three independent guesses: (1/4)^3 = 1/64

Student wrote: 1/4 x 1/4 x 1/4 = 1/64 → correct.

✔ Answer: 1/64

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Now, compiling all final answers clearly:

Final Answer:
1. Independent events: first doesn't affect second; Dependent events: first affects second.
2. 1/6
3. 9/25
4. 1/16
5. 1/25
6. 1/5
7. 1/5
8. 156/625
9. 13/50
10. 1/1650
11. 1/64