Lecture on Probability

What is Probability?

• Definition: Probability measures the likelihood of an event occurring.
• Notation: P(A)
  • Represents the probability of event A occurring.
• Formula: Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes)

Sample Space

• Definition: The set of all possible outcomes that can occur in a situation.
• Example: Flipping a fair coin
  • Possible outcomes: Heads (H) or Tails (T)
  • Sample Space: {H, T}
• Example: Flipping two coins
  • Possible outcomes: HH, HT, TH, TT
  • Determine using a tree diagram.
• Example: Flipping three coins
  • Possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
  • Determine using a tree diagram.
  • Total outcomes: 2^3 = 8

Probability Range

• Range: 0 ≤ P(A) ≤ 1
• Meaning:
  • P(A) = 0: Event cannot happen
  • P(A) = 1: Event will always happen
• Conversion example: P(A) = 0.3 ⟹ 30% chance of occurring (0.3 * 100)

Practical Example

• Example: Probability of people driving a blue car
  • Given: P(A) = 0.20 (20%)
  • Interpretation: Out of 100 people, approximately 20 will drive a blue car.

Problem-Solving with Probability

Example Problems

1. Flipping two coins:

   • Problem: Probability of getting at least one head
   • Sample Space: {HH, HT, TH, TT}
   • Favorable Outcomes: HH, HT, TH
   • Probability: 3/4 = 0.75 (75%)

2. Flipping three coins:

   • Problem: Probability of getting at least two tails

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

   • Favorable Outcomes: HTT, THT, TTH, TTT

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

   • Problem: Probability of getting exactly one tail

   • Favorable Outcomes: HHT, HTH, THH

   • Probability: 3/8 = 0.375 (37.5%)

3. Tossing a Six-Sided Die:

   • Problem: Probability of getting a 2

   • Sample Space: {1, 2, 3, 4, 5, 6}

   • Favorable Outcomes: {2}

   • Probability: 1/6 ≈ 0.167 (16.7%)

   • Problem: Probability of getting a 3 or a 5

   • Favorable Outcomes: {3, 5}

   • Probability: 2/6 = 1/3 ≈ 0.333 (33.3%)

   • Problem: Probability of getting a number at most 4

   • Favorable Outcomes: {1, 2, 3, 4}

   • Probability: 4/6 = 2/3 ≈ 0.667 (66.7%)

   • Problem: Probability of getting a number greater than 3

   • Favorable Outcomes: {4, 5, 6}

   • Probability: 3/6 = 1/2 = 0.5 (50%)

   • Problem: Probability of getting a number less than or equal to 5

   • Favorable Outcomes: {1, 2, 3, 4, 5}

   • Probability: 5/6 ≈ 0.833 (83.3%)

Conclusion

• Now capable of calculating the probability of an event occurring.
• For more resources:
  • Statistics playlist
  • Topics: independent events, dependent events, mutually exclusive events, conditional probability, contingency tables, complementary events
• Search on YouTube: Organic Chemistry Tutor + [Topic]

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