8.6 Examples of Tree Diagrams in Probability

This section provides practical examples illustrating the use of tree diagrams to solve probability problems involving sequential and combined events.

Example 1: Tossing a Coin and Rolling a Die

Scenario: You flip a fair coin once and then roll a standard six-sided die. What is the probability of getting Tails on the coin and an even number on the die?

Step-by-Step Breakdown:

1. Identify Possible Outcomes and Probabilities:

   • Coin Outcomes:
     • Heads (H): Probability = 1/2
     • Tails (T): Probability = 1/2
   • Die Outcomes:
     • Numbers 1, 2, 3, 4, 5, 6: Each has a probability of 1/6.
     • Even Numbers (2, 4, 6): There are 3 favorable outcomes.
     • Probability of rolling an even number = 3/6 = 1/2.

2. Construct the Tree Diagram:

   • Level 1 – Coin Toss:

     • Branch 1: H (Probability: 1/2)
     • Branch 2: T (Probability: 1/2)

   • Level 2 – Rolling the Die (from each coin outcome):

     • From H:
       • 1 (Probability: 1/6)
       • 2 (Probability: 1/6)
       • 3 (Probability: 1/6)
       • 4 (Probability: 1/6)
       • 5 (Probability: 1/6)
       • 6 (Probability: 1/6)
     • From T:
       • 1 (Probability: 1/6)
       • 2 (Probability: 1/6)
       • 3 (Probability: 1/6)
       • 4 (Probability: 1/6)
       • 5 (Probability: 1/6)
       • 6 (Probability: 1/6)

3. Identify the Desired Outcome:

   • We are interested in the path: Tails (T) on the coin AND an Even number (2, 4, or 6) on the die.
   • Probability of getting Tails: $P(Tails) = 1/2$.
   • Probability of getting an even number given that the coin was Tails: $P(\text{Even} | \text{Tails}) = 1/2$.

4. Calculate the Final Probability: To find the probability of both events occurring, we multiply the probabilities along the path of the tree diagram.

   • $P(\text{Tails and Even}) = P(Tails) \times P(\text{Even} | \text{Tails})$
   • $P(\text{Tails and Even}) = 1/2 \times 1/2$
   • $P(\text{Tails and Even}) = 1/4$

Answer: The probability of getting Tails on the coin and an even number on the die is 1/4.

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Example 2: Choosing a Basket and Picking a Fruit

Scenario: You randomly select one of two baskets, Basket A or Basket B.

• Basket A contains apples and oranges in a ratio of 3:7.
• Basket B contains bananas and grapes in an equal ratio of 1:1.

What is the probability of selecting an apple from Basket A AND a grape from Basket B?

Step-by-Step Breakdown:

1. Step 1: Probability of Choosing a Basket:

   • The selection of a basket is random, so each basket has an equal probability of being chosen.
   • $P(\text{Basket A}) = 1/2$
   • $P(\text{Basket B}) = 1/2$

2. Step 2: Probabilities of Fruit within Each Basket:

   • Basket A:
     • The ratio of apples to oranges is 3:7. The total parts are $3 + 7 = 10$.
     • $P(\text{Apple | Basket A}) = 3/10$
     • $P(\text{Orange | Basket A}) = 7/10$
   • Basket B:
     • The ratio of bananas to grapes is 1:1. The total parts are $1 + 1 = 2$.
     • $P(\text{Banana | Basket B}) = 1/2$
     • $P(\text{Grape | Basket B}) = 1/2$

3. Construct the Tree Diagram:

   • Level 1 – Basket Selection:

     • Branch 1: Basket A (Probability: 1/2)
     • Branch 2: Basket B (Probability: 1/2)

   • Level 2 – Fruit Selection (from each basket):

     • From Basket A:
       • Apple (Probability: 3/10)
       • Orange (Probability: 7/10)
     • From Basket B:
       • Banana (Probability: 1/2)
       • Grape (Probability: 1/2)

4. Calculate Probabilities for Specific Events:

   • Probability of selecting an apple from Basket A:

     • This involves the path: Basket A $\rightarrow$ Apple.
     • $P(\text{Apple from A}) = P(\text{Basket A}) \times P(\text{Apple | Basket A})$
     • $P(\text{Apple from A}) = 1/2 \times 3/10 = 3/20$

   • Probability of selecting a grape from Basket B:

     • This involves the path: Basket B $\rightarrow$ Grape.
     • $P(\text{Grape from B}) = P(\text{Basket B}) \times P(\text{Grape | Basket B})$
     • $P(\text{Grape from B}) = 1/2 \times 1/2 = 1/4$

5. Calculate the Combined Probability:

   • We want the probability of selecting an apple from Basket A AND a grape from Basket B. Since these are independent sequential events, we multiply their individual probabilities.
   • $P(\text{Apple from A and Grape from B}) = P(\text{Apple from A}) \times P(\text{Grape from B})$
   • $P(\text{Apple from A and Grape from B}) = 3/20 \times 1/4$
   • $P(\text{Apple from A and Grape from B}) = 3/80$

Answer:

• The probability of selecting an apple from Basket A is 3/20.
• The probability of selecting a grape from Basket B is 1/4.
• The probability of both events happening together (apple from A AND grape from B) is 3/80.

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Key Concepts Illustrated:

• Multi-stage Probability: Problems involving a sequence of events.
• Independent Events: The outcome of one event does not affect the outcome of another (e.g., coin toss and die roll are independent).
• Conditional Probability: The probability of an event occurring given that another event has already occurred (e.g., probability of picking a specific fruit given that you chose a particular basket).
• Multiplication Rule: For independent events A and B, $P(A \text{ and } B) = P(A) \times P(B)$. For sequential events where the second depends on the first, $P(A \text{ and } B) = P(A) \times P(B|A)$.
• Sum of Probabilities: Probabilities along branches originating from a single node must sum to 1 (e.g., P(Heads) + P(Tails) = 1; P(Apple|A) + P(Orange|A) = 1).

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Potential Interview Questions:

• What is the probability of getting Tails on a coin and an even number on a die? How do you calculate it using a tree diagram?
• Explain how a tree diagram can be used to solve combined event probability problems.
• What is conditional probability? Give an example involving fruit baskets.
• How do you calculate the probability of two independent events occurring together?
• What is the difference between independent and dependent events in probability?
• How would you model the problem of selecting fruits from different baskets using probability concepts?
• Can you explain the multiplication rule in the context of sequential probability events?
• Why is it important that probabilities from a single node sum to 1 in a tree diagram?
• How do you handle multiple stages in probability problems?
• Describe a real-life scenario where you would apply a probability tree diagram.