Peluang (Part 1) | Definisi Peluang, Komplemen Kejadian dan Frekuensi Harapan Matematika Kelas 12

Introduction

In this tutorial, we will explore the fundamental concepts of probability, including definitions, complementary events, and expected frequency, suitable for grade 12 mathematics. Understanding these concepts is crucial for grasping more complex statistical ideas and solving real-world problems.

Step 1: Understanding Probability

• Definition of Probability: Probability measures the likelihood of an event occurring, expressed as a number between 0 and 1.
  • An event with a probability of 0 will not occur.
  • An event with a probability of 1 is certain to occur.
• Formula:
  • The probability of an event ( A ) can be calculated using the formula: [ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} ]

Step 2: Exploring Examples of Probability

• Example 1: Calculating the probability of rolling a specific number on a six-sided die.

  • Favorable outcomes (rolling a 3): 1
  • Total possible outcomes: 6
  • Thus, ( P(3) = \frac{1}{6} ).

• Example 2: Finding the probability of drawing a heart from a standard deck of cards.

  • Favorable outcomes (hearts): 13
  • Total possible outcomes: 52
  • Therefore, ( P(\text{heart}) = \frac{13}{52} = \frac{1}{4} ).

Step 3: Complementary Events

• Definition of Complementary Events: The complement of an event ( A ) is the event that ( A ) does not occur.
• Formula:
  • The probability of the complement of event ( A ) is given by: [ P(A') = 1 - P(A) ]

Step 4: Examples of Complementary Events

• Example 1: If ( P(A) = \frac{1}{6} ) (rolling a 3), then the probability of not rolling a 3 is: [ P(A') = 1 - \frac{1}{6} = \frac{5}{6} ]

• Example 2: If you have a 25% chance of rain (event ( A )), the chance of no rain (event ( A' )) is: [ P(A') = 1 - 0.25 = 0.75 ]

Step 5: Expected Frequency

• Definition: Expected frequency (or mathematical expectation) is the predicted number of times an event will occur in a large number of trials.
• Formula:
  • Expected frequency can be calculated as: [ E = P(A) \times N ]
  • Where ( N ) is the total number of trials.

Step 6: Examples of Expected Frequency

• Example 1: If you conduct 100 trials and the probability of an event ( A ) is ( \frac{1}{4} ): [ E = \frac{1}{4} \times 100 = 25 ]

  • You can expect the event ( A ) to occur 25 times.

• Example 2: For a probability of rolling a 2 on a die (( P = \frac{1}{6} )): [ E = \frac{1}{6} \times 120 = 20 ]

  • Thus, you can expect to roll a 2 about 20 times in 120 rolls.

Conclusion

In this tutorial, we covered the basics of probability, complementary events, and expected frequency. Understanding these concepts is essential for solving more advanced problems in statistics and mathematics. For further learning, consider reviewing prerequisite materials on counting principles and exploring additional examples to reinforce your understanding.