Probability Tree Diagrams

Overview

• Probability tree diagrams are visual tools to represent possible outcomes of a sequence of events.
• Useful for determining probabilities in games or scenarios involving multiple events.

Example 1: Spinner Game

• Scenario: Sam plays a game twice using a spinner.
  • Outcomes: Win (W) or Lose (L).
• Possible Outcomes:
  • Win-Win (W-W)
  • Win-Lose (W-L)
  • Lose-Win (L-W)
  • Lose-Lose (L-L)
• Tree Diagram Construction:
  • First event: Draw branches for Win or Lose.
  • Second event: Draw additional branches for each first event outcome.
  • Resulting branches represent all possible outcomes.

Example 2: Dice and Coin Game

• Scenario: Ria rolls a fair six-sided dice and then throws a fair coin.
  • Wins by rolling a six followed by tails.
• Tree Diagram Construction:
  • First event (Dice): Outcomes are rolling a six or not.
    • Probability of six: (\frac{1}{6})
    • Probability of not-six: (\frac{5}{6})
  • Second event (Coin): Outcomes are heads or tails for each prior event.
    • Probability each: (\frac{1}{2})
• Calculating Probabilities:
  • Probability of winning (6, T): Multiply probabilities along the path: (\frac{1}{6} \times \frac{1}{2} = \frac{1}{12})

Example 3: Counters in a Bag

• Scenario: A bag with 3 green and 5 red counters, replaced after being drawn.
• Tree Diagram Structure:
  • First draw: Green or Red
  • Second draw: Green or Red
  • Probabilities:
    • Green: (\frac{3}{8})
    • Red: (\frac{5}{8})
• Example Calculations:
  • Probability both red: (\frac{5}{8} \times \frac{5}{8} = \frac{25}{64})
  • Probability one red: Different paths such as Green-Red or Red-Green, add probabilities of different paths.

Example 4: Tests with Decimals

• Scenario: Luke takes a math test and an English test.
  • Math pass probability: 0.8
  • English pass probability: 0.7
• Tree Diagram Considerations:
  • Probabilities need to be added for 'fail'.
  • Probabilities add up to 1 for each event.
• Example Calculations:
  • Probability of passing both: Multiply along path of passing both.
  • Probability of failing both: Multiply along path of failing both.
  • Using "at least one pass": Consider all paths where one or more tests are passed.

Example 5: Altered Game Scenario

• Scenario: Lauren rolls a dice and wins if she rolls an even number on the second roll, loses if she rolls a five on the first.
• Tree Diagram Insights:
  • Remove second roll branches if first roll is a 5.
  • Probabilities for the first roll of not a five and a five.
  • Second roll outcomes: Even or Odd.
• Example Calculations:
  • Probability of winning: Not a five followed by an even number.
  • Probability of losing: Either instantly by rolling a five or by not rolling an even number after not rolling a five.

Summary

• Probability tree diagrams help visualize and calculate complex probability scenarios.
• Key skill: Multiplying probabilities along branches for a particular path.
• In cases with multiple routes, add probabilities to find overall probability.