what are independent and dependent events

what are independent and dependent events

Independent events are events that are not affected by prior events or outcomes. For example, rolling a dice and flipping a coin are independent events because the outcome of one event does not affect the outcome of the other.

Dependent events are events that are affected by prior events or outcomes. For example, drawing two cards from a deck of cards and the outcome of the first card will affect the outcome of the second card.

how to know if events are dependent of independent in probability prove by formula and numerical question.

To determine if two events are independent or dependent in probability, we can use the following rules and formulas:

Independence Rule:

Two events AA and BB are independent if the probability of both events occurring together is equal to the product of their individual probabilities. Mathematically, this is written as: P(A∩B)=P(A)×P(B)P(A \cap B) = P(A) \times P(B)

If this equation holds, the events are independent. If it does not hold, the events are dependent.

Dependence Rule:

Two events AA and BB are dependent if the probability of one event occurring is affected by whether the other event has occurred. This can be proven using conditional probability, which is the probability of event AA occurring given that event BB has already occurred, written as P(A∣B)P(A|B).

If the following equation holds: P(A∣B)=P(A)P(A|B) = P(A)

Then the events are independent. If this is not the case, the events are dependent.

Example 1: Independent Events

Let’s consider two independent events:

• Event A: Rolling a dice and getting a 3, P(A)=16P(A) = \frac{1}{6}
• Event B: Flipping a coin and getting heads, P(B)=12P(B) = \frac{1}{2}

The probability of both events occurring together is: P(A∩B)=P(A)×P(B)=16×12=112P(A \cap B) = P(A) \times P(B) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}

Since the probability of both events occurring together is equal to the product of their individual probabilities, the events are independent.

Example 2: Dependent Events

Let’s consider two dependent events:

• Event A: Drawing a card from a deck and getting a red card (hearts or diamonds), P(A)=2652=12P(A) = \frac{26}{52} = \frac{1}{2}
• Event B: Drawing a second card from the deck without replacement and getting another red card.

After drawing the first red card, the probability of drawing another red card is now P(B∣A)=2551P(B|A) = \frac{25}{51}, since one red card has been removed from the deck. The probability of drawing two red cards together is: P(A∩B)=P(A)×P(B∣A)=12×2551=25102P(A \cap B) = P(A) \times P(B|A) = \frac{1}{2} \times \frac{25}{51} = \frac{25}{102}

This probability is not equal to the product of the individual probabilities of the two events occurring independently. Therefore, these events are dependent.

Conclusion:

• Independent events: The probability of both events occurring together equals the product of their individual probabilities.
• Dependent events: The probability of both events occurring together does not equal the product of their individual probabilities, and the outcome of one event influences the probability of the other event.

Formulas:

• Independent Events: P(A∩B)=P(A)×P(B)P(A \cap B) = P(A) \times P(B)
• Dependent Events: P(A∩B)=P(A)×P(B∣A)P(A \cap B) = P(A) \times P(B|A)