(5) Introduction to Probability, Basic Overview - Sample Space, & Tree Diagrams

Introduction to Probability

• Definition: Probability measures the likelihood of an event occurring.

• Notation: P(A) denotes the probability of event A occurring.

• Formula: Probability of an event = (Number of favorable outcomes) / (Total possible outcomes).

Sample Space

• Sample Space: Set of all possible outcomes of an event.

• Example (Flipping a Coin):

  • Outcomes: Heads (H), Tails (T).

  • Sample Space: {H, T}.

Example with Two Coins

• Outcomes when flipping two coins:

  • Tree Diagram illustrates possibilities:

    • First coin: Heads (H) or Tails (T).

    • If H on first: second can be H or T.

    • If T on first: second can also be H or T.

  • Possible outcomes:

    • HH (Heads, Heads)

    • HT (Heads, Tails)

    • TH (Tails, Heads)

    • TT (Tails, Tails)

  • Sample Space for two coins: {HH, HT, TH, TT}.

Example with Three Coins

• Sample Space of Flipping Three Coins:

  • Follow the tree diagram strategy:

    • First coin: H or T.

    • Second coin: H or T for each first coin outcome.

    • Third coin: H or T for each second coin outcome.

  • Outcomes include:

    • HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

  • Total outcomes = 2^3 = 8.

Calculating Probability

• Probability Range: Always between 0 (impossible event) and 1 (certain event).

• Example Calculation:

  • If P(A) = 0.3, then there's a 30% chance of A occurring.

Practical Examples

Blue Car Probability

• Example of Blue Cars in a Population:

  • P(Blue Car) = 0.20

  • Out of 100 people, expect around 20 to drive a blue car.

Flipping Coins

1. At Least One Head with Two Coins:

   • Sample Space: {HH, HT, TH, TT}.

   • Favorable Outcomes: HH, HT, TH (3 outcomes).

   • Probability = 3/4 = 0.75 (75%).

2. At Least Two Tails with Three Coins:

   • Sample Space: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

   • Favorable Outcomes: HTT, THT, TTH, TTT (4 outcomes).

   • Probability = 4/8 = 1/2 = 0.5 (50%).

3. Exactly One Tail with Three Coins:

   • Outcomes: HHT, HTH, THH. (3 outcomes).

   • Probability = 3/8 = 0.375 (37.5%).

Tossing a Die

• Example with a Six-Sided Die:

1. Probability of Getting a 2:

   • Sample Space: {1, 2, 3, 4, 5, 6}.

   • Probability = 1/6 = 0.167 (approximately 16.7%).

2. Probability of Getting a 3 or 5:

   • Favorable Outcomes: 3, 5 (2 outcomes).

   • Probability = 2/6 = 1/3 = 0.333 (approximately 33.3%).

3. Probability of Numbers at Most 4:

   • Outcomes: {1, 2, 3, 4} (4 outcomes).

   • Probability = 4/6 = 2/3 (approximately 66.7%).

4. Probability of Numbers Greater than 3:

   • Outcomes: {4, 5, 6} (3 outcomes).

   • Probability = 3/6 = 1/2 (50%).

5. Probability of Numbers Less than or Equal to 5:

   • Outcomes: {1, 2, 3, 4, 5} (5 outcomes).

   • Probability = 5/6 (approximately 83.3%).

Conclusion

• Understanding probability involves calculating the likelihood of events based on favorable outcomes relative to total outcomes.

• Practical application can help reinforce the concept.

• Further resources available for advanced topics in probability.