Probability Tree Diagrams

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🌳 Part 1: Building Probability Trees

1. A bag contains 4 green marbles and 3 yellow marbles. Two marbles are drawn without replacement. Complete the probability tree diagram by filling in the missing probabilities:

2. Using the tree diagram from Question 1, calculate the following probabilities:

a) P(both marbles are green) = _____________

b) P(first marble is green AND second marble is yellow) = _____________

c) P(at least one marble is yellow) = _____________

3. A coin is flipped twice. Which tree diagram correctly shows this situation?

The probabilities change after the first flip

Each branch shows probability of 1/2

The total number of outcomes is 3

The second flip depends on the first flip

🎯 Part 2: With and Without Replacement

4. A deck has 5 red cards and 3 black cards. Compare these two scenarios by drawing tree diagrams and calculating probabilities:

Scenario A: Two cards drawn with replacement

P(both red) = _____________

Scenario B: Two cards drawn without replacement

P(both red) = _____________

5. Which statement about replacement is correct?

With replacement gives higher probabilities for matching outcomes

Without replacement means probabilities stay the same

Replacement doesn't affect the final probabilities

Without replacement makes events independent

🤔 Part 3: Critical Thinking

6. Sarah claims: "When flipping a coin twice, there are three possible outcomes: two heads, two tails, or one of each. So each outcome has probability 1/3." Use a probability tree to explain why Sarah's reasoning is incorrect.

7. A weather app claims there's a 70% chance of rain today and 60% chance tomorrow. It states: "The probability of rain on both days is 42%." What assumption is the app making? Is this assumption reasonable?

8. You conduct an experiment drawing coloured balls from a bag 50 times. Your experimental results show P(red) = 0.32, but your theoretical calculation gives P(red) = 0.4. Give two possible explanations for this difference:

Explanation 1: ________________________________________________

Explanation 2: ________________________________________________