CodeForCrack - Independent and Mutually Exclusive Events

In probability theory, understanding the relationship between events is crucial. Two key concepts that come up often are independent events and mutually exclusive events. These concepts help in calculating the probabilities of different combinations of events and understanding how they relate to each other.

1. Independent Events

Two events are said to be independent if the occurrence of one event does not affect the occurrence of the other. In other words, the probability of both events occurring together is the product of their individual probabilities. Mathematically, this can be written as:

If P(A) is the probability of event A occurring, and P(B) is the probability of event B occurring, then events A and B are independent if:

P(A ∩ B) = P(A) * P(B)

Example: Tossing a coin twice. The outcome of the first toss (heads or tails) does not affect the outcome of the second toss. Therefore, these two events are independent.
Example: P(Heads on toss 1) = 0.5, P(Heads on toss 2) = 0.5, and the probability of both heads occurring is: P(Heads on toss 1 ∩ Heads on toss 2) = 0.5 * 0.5 = 0.25.

2. Mutually Exclusive Events

Two events are said to be mutually exclusive if they cannot occur at the same time. In other words, the occurrence of one event excludes the possibility of the other event occurring. The probability of both events occurring together is zero.

Mathematically, this can be written as:

P(A ∩ B) = 0

Example: When rolling a six-sided die, the event of getting an even number and the event of getting an odd number are mutually exclusive. You cannot roll an even and an odd number at the same time.
Example: P(Even) = {2, 4, 6}, P(Odd) = {1, 3, 5}, and the probability of both events occurring is P(Even ∩ Odd) = 0.

3. Calculating Probabilities

Understanding how to calculate probabilities for independent and mutually exclusive events is essential in probability theory.

For Independent Events:

For independent events, the probability of both events occurring is simply the product of the probabilities of the individual events.

Example: If P(A) = 0.4 and P(B) = 0.6 and A and B are independent, then:
P(A ∩ B) = P(A) * P(B) = 0.4 * 0.6 = 0.24

For Mutually Exclusive Events:

For mutually exclusive events, the probability of either event occurring is the sum of their individual probabilities.

Example: If P(A) = 0.5 and P(B) = 0.3 and A and B are mutually exclusive, then:
P(A ∪ B) = P(A) + P(B) = 0.5 + 0.3 = 0.8