Probability worksheet featuring a marble selection problem with various probability questions and event type identification.

Let's solve each problem step by step using the information provided.

Given:

- Blue marbles: 4
- Red marbles: 5
- Green marble: 1
- Black marbles: 2

Total marbles = 4 + 5 + 1 + 2 = 12

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1. P(black)

We are selecting one marble at random.
Probability of selecting a black marble:

$$
P(\text{black}) = \frac{\text{Number of black marbles}}{\text{Total marbles}} = \frac{2}{12} = \frac{1}{6}
$$

✔ Answer: $ \frac{1}{6} $

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2. P(black or blue)

This is the probability of selecting either a black or a blue marble.

Since these are mutually exclusive (a marble can't be both black and blue), we add the probabilities:

$$
P(\text{black or blue}) = P(\text{black}) + P(\text{blue}) = \frac{2}{12} + \frac{4}{12} = \frac{6}{12} = \frac{1}{2}
$$

✔ Answer: $ \frac{1}{2} $

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3. P(black and then blue); put the marbles back after each selection

This means we are doing two independent events with replacement.

- First, pick a black marble: $ P(\text{black}) = \frac{2}{12} $
- Then, put it back, so total remains 12.
- Pick a blue marble: $ P(\text{blue}) = \frac{4}{12} $

Since the selections are independent:

$$
P(\text{black and then blue}) = P(\text{black}) \times P(\text{blue}) = \frac{2}{12} \times \frac{4}{12} = \frac{8}{144} = \frac{1}{18}
$$

✔ Answer: $ \frac{1}{18} $

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4. P(black and then blue); don’t put the marbles back

Now, this is without replacement, so the events are dependent.

- First, pick a black marble: $ P(\text{black}) = \frac{2}{12} $
- After removing one black marble, total marbles left = 11, black marbles left = 1
- Now pick a blue marble: $ P(\text{blue | black already picked}) = \frac{4}{11} $

So,

$$
P(\text{black and then blue}) = \frac{2}{12} \times \frac{4}{11} = \frac{8}{132} = \frac{2}{33}
$$

✔ Answer: $ \frac{2}{33} $

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Question: What type of probability situation?

Let’s analyze each:

1. P(black) → Selecting one marble. This is a single event.
✔ Single event

2. P(black or blue) → Two outcomes that cannot happen at the same time. They are mutually exclusive.
✔ Two mutually exclusive events

3. P(black and then blue) with replacement → Two separate draws, outcomes don’t affect each other.
✔ Multiple independent events

4. P(black and then blue) without replacement → The first draw affects the second.
✔ Multiple dependent events

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✔ Final Answers:

1. $ \frac{1}{6} $ — Single event
2. $ \frac{1}{2} $ — Two mutually exclusive events
3. $ \frac{1}{18} $ — Multiple independent events
4. $ \frac{2}{33} $ — Multiple dependent events

Let me know if you'd like a visual explanation or diagram!