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Understanding the Probability of Events and Their Calculations

What Is Probability?

Probability measures the likelihood of an event occurring. It is expressed as a value between 0 and 1, where 0 means the event will not occur, and 1 means the event is certain to happen. The closer the value is to 1, the more likely the event will take place. In its basic form, probability is calculated by dividing the number of favorable outcomes by the total number of outcomes.

Events can be independent, mutually exclusive, or conditional, and these properties affect how probabilities are calculated.

Complement of an Event

The complement of event A, written as P(A’), is the probability that event A does not happen. If P(A) is the probability of event A, then:

P(A’) = 1 – P(A)

For example, if P(A) = 0.65, then:
P(A’) = 1 – 0.65 = 0.35

So, there is a 35% chance that event A does not occur.

This applies similarly to any event B:
P(B’) = 1 – P(B)

Intersection of Events (AND)

The intersection of two events A and B, denoted as P(A ∩ B), is the probability that both events happen. If events A and B are independent, then:

P(A ∩ B) = P(A) × P(B)

For example, rolling a 6 on two dice rolls:
P(A) = 1/6, P(B) = 1/6
P(A ∩ B) = 1/6 × 1/6 = 1/36

If the events are dependent, you must use conditional probability:

P(A ∩ B) = P(A) × P(B|A)

Example:

• P(A) = 3/10 (drawing a blue marble)
• P(B|A) = 7/9 (drawing a black marble after drawing blue without replacement)
• P(A ∩ B) = 3/10 × 7/9 = 0.2333

Union of A and B

The union of events A and B, written as P(A ∪ B), is the probability that at least one of the events occurs.

For mutually exclusive events:
P(A ∪ B) = P(A) + P(B)

For non-mutually exclusive events:
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Example:
Rolling a dice:

• P(even) = 3/6
• P(multiple of 3) = 2/6
• P(even and multiple of 3) = 1/6
• P(A ∪ B) = 3/6 + 2/6 – 1/6 = 2/3

Exclusive OR of A and B

P(A XOR B) is the probability that either A or B occurs, but not both.

P(A XOR B) = P(A) + P(B) – 2 × P(A ∩ B)

Example:

• P(Snickers) = 0.65
• P(Reese’s) = 0.349
• P(Both) = 0.001

P(A XOR B) = 0.65 + 0.349 – 2 × 0.001 = 0.997

Normal Distribution
The normal (or Gaussian) distribution is a continuous probability distribution described by the formula:

f(x) = (1 / √(2πσ^2)) × e^{-((x – μ)^2 / 2σ^2)}

Where:

• μ = mean
• σ = standard deviation

If μ = 0 and σ = 1, it’s a standard normal distribution.

Example:
Find the probability a student is between 60 and 72 inches tall given:

• μ = 68 inches
• σ = 4 inches

Standardize:

• Z1 = (60 – 68)/4 = -2
• Z2 = (72 – 68)/4 = 1

From Z-table:

• P(0 to 2) = 0.47725
• P(0 to 1) = 0.34134

P(-2 to 1) = 0.47725 + 0.34134 = 0.81859 (81.859%)

Z-Table Reference (0 to Z)
Use a Z-table to find the area under the standard normal curve. The table provides values for the probability that a standard normal variable falls between 0 and a positive Z value. For negative Z-values, use symmetry.

Examples:

• Z = 1.0 → P = 0.34134
• Z = 2.0 → P = 0.47725
• Z = 3.0 → P = 0.49865

Use these values when calculating areas between points under a normal curve or for confidence intervals.

Related Calculators:
Sample Size Calculator, Number Sequence Calculator

External Resources:
Probability Calculator on Calculator.net