Probability Year 10 Notes

Probability Terminology

• Definition of Probability:

  • Measures the chance of an event occurring.
  • Written as P(result).

• Event:

  • A measurable test or observation with various outcomes.

• Outcomes:

  • Results of an event; successes are often of interest.

• Mutually Exclusive Events:

  • Events that cannot happen at the same time.
  • Their probabilities add up to 1.
  • Notation: P(A) + P(A') = 1, where A' is the complement of A.

• Probability Values:

  • Range: 0 (impossible) to 1 (certain).
  • Common expressions: 0% to 100% (0 to 1).
  • Avoid vague terms like "likely"; instead, provide precise probabilities.

• Useful Notation:

  • Prefer fractions or decimals for calculations; percentages can complicate math.

Calculating Simple Probability

• Experimental Probability:

  • Based on conducting trials:

  Probability = Number of favorable outcomes / Total number of trials.

  • It is an estimate and may vary with each trial.

• Theoretical Probability:

  • Probability calculated assuming each outcome is equally likely:

  Probability = Number of favorable outcomes / Total number of possible outcomes.

  • Must ensure outcomes are divided equally for accurate calculations.

• Sampling Considerations:

  • Sample should be:
  • Random: Each member has an equal chance of being selected.
  • Independent: One outcome does not affect another.
  • Sufficiently Large: The larger the sample, the more reliable the results.

Expected Value

• Definition:

  • The expected value calculates the average number of successes over a number of trials:

  Expected Value = Probability × Number of Trials.

• Example:

  • Die roll: Probability of rolling a 6 = 1/6.
  • If rolled 20 times, expected value = 1/6 × 20 = 3.33, rounded to 3 since you can't have a fraction of a success.

Probability Calculations

• Adding Probabilities:

  • For mutually exclusive events A and B:

  P(A or B) = P(A) + P(B).

• Multiplying Probabilities:

  • For independent events occurring in succession:

  P(A then B) = P(A) × P(B).

• Real-Life Applications:

  • Consider all arrangements giving the same outcome as different ways.
  • Simplify complex scenarios by ignoring unimportant variations (e.g., different suits in cards).

Probability Trees

• Building a Probability Tree:

  • Identify various potential outcomes at each event.
  • Start with the first event and draw branches to possible outcomes.
  • Continue adding branches for subsequent events.
  • Label each branch with its probability.
  • Sum probabilities of paths leading to the outcome of interest.

• Tree Structure:

  • Each vertical row represents a separate event; each branch denotes possible outcomes.
  • Important to ensure clarity in execution:
  • Maintain separate branching for simultaneous versus sequential events.

• Example in Trees:

  • If the first event has outcomes Success (S) and Failure (F):
  • P(S, S) = 0.4 × 0.4 = 0.16
  • P(S, F) = 0.4 × 0.6 = 0.24
  • P(F, S) = 0.6 × 0.4 = 0.24
  • P(F, F) = 0.6 × 0.6 = 0.36

Sampling Without Replacement

• Concept:

  • Each choice alters the pool for subsequent selections, affecting probabilities.
  • Probabilities must be recalculated for each selection:

  Probability = Number of favorable outcomes / Adjusted total outcomes.

• Example:

  • Given 10 cards (5 red):
  • P(1st is red) = 5/10
  • P(2nd is red) = 4/9 (one successful outcome removed)
  • P(2 reds) = P(1st is red) × P(2nd is red) = (5/10) × (4/9) = 0.2.

• Mixed Results Probability:

  • Calculate probabilities for different colors drawn from the same group, considering each step separately to ensure accuracy.