Probability – Addition rule of Probability using Axiomatic definition

  • Probability is a branch of mathematics that deals with the likelihood of events occurring.
  • This topic focuses on the addition rule of probability using the axiomatic definition.
  • The axiomatic definition states that the probability of the union of two or more mutually exclusive events is the sum of their individual probabilities.
  • Let’s understand this concept with an example.

Example 1:

  • Consider a deck of playing cards.

  • What is the probability of drawing a red card or a face card? Solution:

  • We have two mutually exclusive events: drawing a red card and drawing a face card.

  • The probability of drawing a red card is 26/52 or 1/2.

  • The probability of drawing a face card is 12/52 or 3/13.

  • By applying the addition rule of probability, the probability of drawing a red card or a face card is (1/2) + (3/13) = 11/26.

  • The addition rule of probability can be extended to more than two events as well.

  • Let’s understand this concept further with another example. Example 2:

  • Consider rolling a fair six-sided die.

  • What is the probability of getting an odd number or a number greater than 4? Solution:

  • We have two mutually exclusive events: getting an odd number and getting a number greater than 4.

  • The probability of getting an odd number is 3/6 or 1/2.

  • The probability of getting a number greater than 4 is 2/6 or 1/3.

  • By applying the addition rule of probability, the probability of getting an odd number or a number greater than 4 is (1/2) + (1/3) = 5/6.

  • In some cases, the events may not be mutually exclusive.

  • Let’s explore the addition rule of probability for non-mutually exclusive events. Example 3:

  • Consider drawing a card from a deck.

  • What is the probability of drawing a heart or a face card? Solution:

  • The events “drawing a heart” and “drawing a face card” are not mutually exclusive, as the card can be both a heart and a face card.

  • We need to consider the overlap between these events.

  • The probability of drawing a heart is 13/52 or 1/4.

  • The probability of drawing a face card is 12/52 or 3/13.

  • To find the probability of drawing a heart or a face card, we subtract the probability of drawing a heart and a face card (which is also double-counted) from the sum of individual probabilities.

  • Therefore, the probability of drawing a heart or a face card is (1/4) + (3/13) - (1/26) = 15/26.

  • The addition rule of probability using the axiomatic definition is a fundamental concept in probability theory.

  • It helps us calculate the probability of events occurring together or separately.

  • It is applicable to mutually exclusive events as well as non-mutually exclusive events.

  • Understanding this concept is crucial for solving various probability problems in mathematics and real-life scenarios.

  • Let’s practice more examples to strengthen our understanding.

Probability – Addition rule of Probability using Axiomatic definition

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Probability – Addition rule of Probability using Axiomatic definition Probability is a branch of mathematics that deals with the likelihood of events occurring. This topic focuses on the addition rule of probability using the axiomatic definition. The axiomatic definition states that the probability of the union of two or more mutually exclusive events is the sum of their individual probabilities. Let’s understand this concept with an example. Example 1: Consider a deck of playing cards. What is the probability of drawing a red card or a face card? Solution: We have two mutually exclusive events: drawing a red card and drawing a face card. The probability of drawing a red card is 26/52 or 1/2. The probability of drawing a face card is 12/52 or 3/13. By applying the addition rule of probability, the probability of drawing a red card or a face card is (1/2) + (3/13) = 11/26. The addition rule of probability can be extended to more than two events as well. Let’s understand this concept further with another example. Example 2: Consider rolling a fair six-sided die. What is the probability of getting an odd number or a number greater than 4? Solution: We have two mutually exclusive events: getting an odd number and getting a number greater than 4. The probability of getting an odd number is 3/6 or 1/2. The probability of getting a number greater than 4 is 2/6 or 1/3. By applying the addition rule of probability, the probability of getting an odd number or a number greater than 4 is (1/2) + (1/3) = 5/6. In some cases, the events may not be mutually exclusive. Let’s explore the addition rule of probability for non-mutually exclusive events. Example 3: Consider drawing a card from a deck. What is the probability of drawing a heart or a face card? Solution: The events “drawing a heart” and “drawing a face card” are not mutually exclusive, as the card can be both a heart and a face card. We need to consider the overlap between these events. The probability of drawing a heart is 13/52 or 1/4. The probability of drawing a face card is 12/52 or 3/13. To find the probability of drawing a heart or a face card, we subtract the probability of drawing a heart and a face card (which is also double-counted) from the sum of individual probabilities. Therefore, the probability of drawing a heart or a face card is (1/4) + (3/13) - (1/26) = 15/26. The addition rule of probability using the axiomatic definition is a fundamental concept in probability theory. It helps us calculate the probability of events occurring together or separately. It is applicable to mutually exclusive events as well as non-mutually exclusive events. Understanding this concept is crucial for solving various probability problems in mathematics and real-life scenarios. Let’s practice more examples to strengthen our understanding. Probability – Addition rule of Probability using Axiomatic definition