Probability

Probability - Addition Rule

The empirical definition of probability is based on the relative frequency of an event occurring in a large number of trials.
Drawback 1: It requires a large number of trials to achieve accurate results. This can be impractical or time-consuming in real-life scenarios.
Drawback 2: It assumes that all trials are independent and identically distributed. In reality, many events are dependent on each other.
Drawback 3: It does not provide a theoretical foundation for probability. It is merely an approximation based on observations.
Drawback 4: It cannot be used to calculate the probability of events that have not been observed before.
Drawback 5: It does not account for subjective beliefs or personal judgments in the determination of probability.

The classical definition of probability is based on the assumption of equally likely outcomes in a sample space.
It is applicable only to situations where all outcomes are equally likely.
The probability of an event A is given by the ratio of the number of favorable outcomes to the total number of outcomes in the sample space.
Formula: P(A) = N(A) / N(S)
N(A) represents the number of favorable outcomes for event A.
N(S) represents the total number of outcomes in the sample space.
Example: Tossing a fair coin has two equally likely outcomes - heads or tails. The probability of getting heads is 1/2, and the probability of getting tails is also 1/2.

The relative frequency definition of probability is based on the relative frequency of an event occurring in a long-run series of observations.
It assumes that the probability of an event is equal to the limit of the relative frequency of that event.
The probability of an event A is calculated by dividing the number of times event A occurs by the total number of trials.
Formula: P(A) = lim (n -> ∞) N(A) / n
Example: Rolling a fair six-sided die. After rolling the die 600 times, if we observe that 100 times the number 3 comes up, then the relative frequency of getting 3 is 100/600, which can be considered as the probability of getting a 3.

The axiomatic definition of probability is based on a set of mathematical axioms and rules.
It provides a theoretical foundation for probability and is widely used in probability theory.
The probability of an event A is a non-negative real number that satisfies the following three axioms:
1. Axiom of Non-Negativity: P(A) ≥ 0 for any event A.
2. Axiom of Unit Measure: P(S) = 1, where S is the sample space.
3. Axiom of Countable Additivity: For any countable sequence of mutually exclusive events A_1, A_2, …, P(A_1 ∪ A_2 ∪ …) = P(A_1) + P(A_2) + …
The axiomatic definition allows for more flexibility and generalizations compared to other definitions.
Example: The probability of rolling a fair six-sided die and getting an odd number is 1/2.

The empirical definition of probability is based on the relative frequency of an event occurring in a large number of trials.
Drawback 1: It requires a large number of trials to achieve accurate results. This can be impractical or time-consuming in real-life scenarios.
Drawback 2: It assumes that all trials are independent and identically distributed. In reality, many events are dependent on each other.
Drawback 3: It does not provide a theoretical foundation for probability. It is merely an approximation based on observations.
Drawback 4: It cannot be used to calculate the probability of events that have not been observed before.
Drawback 5: It does not account for subjective beliefs or personal judgments in the determination of probability.

The classical definition of probability is based on the assumption of equally likely outcomes in a sample space.
It is applicable only to situations where all outcomes are equally likely.
The probability of an event A is given by the ratio of the number of favorable outcomes to the total number of outcomes in the sample space.
Formula: P(A) = N(A) / N(S)
N(A) represents the number of favorable outcomes for event A.
N(S) represents the total number of outcomes in the sample space.
Example: Tossing a fair coin has two equally likely outcomes - heads or tails. The probability of getting heads is 1/2, and the probability of getting tails is also 1/2.

The relative frequency definition of probability is based on the relative frequency of an event occurring in a long-run series of observations.
It assumes that the probability of an event is equal to the limit of the relative frequency of that event.
The probability of an event A is calculated by dividing the number of times event A occurs by the total number of trials.
Formula: P(A) = lim (n -> ∞) N(A) / n
Example: Rolling a fair six-sided die. After rolling the die 600 times, if we observe that 100 times the number 3 comes up, then the relative frequency of getting 3 is 100/600, which can be considered as the probability of getting a 3.

The axiomatic definition of probability is based on a set of mathematical axioms and rules.
It provides a theoretical foundation for probability and is widely used in probability theory.
The probability of an event A is a non-negative real number that satisfies the following three axioms:
1. Axiom of Non-Negativity: P(A) ≥ 0 for any event A.
2. Axiom of Unit Measure: P(S) = 1, where S is the sample space.
3. Axiom of Countable Additivity: For any countable sequence of mutually exclusive events A_1, A_2, …, P(A_1 ∪ A_2 ∪ …) = P(A_1) + P(A_2) + …
The axiomatic definition allows for more flexibility and generalizations compared to other definitions.
Example: The probability of rolling a fair six-sided die and getting an odd number is 1/2.