Conditional Probability Explained

Conditional Probability

• Conditional probability deals with the probability of an event B occurring, given that another event A has already occurred.
• Notation: $P(B|A)$, which reads as "the probability of B given A". The vertical bar means "given that" or "knowing that" A has already happened.

Understanding Conditional Probability

• If A has already happened, then event A becomes the new sample space.
• The probability of B given A is the ratio of the intersection of A and B to the probability of A.
• Formula: $P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$
• Where:
  • $P(B|A)$ is the conditional probability of B given A.
  • $P(A \text{ and } B)$ is the probability of both A and B occurring.
  • $P(A)$ is the probability of A occurring.

Example: Job Applicants

• Consider a company with job applicants who either have experience or don't, and either have a degree or don't.
• Four categories of applicants:
  • Has a degree and experience.
  • Has a degree but no experience.
  • Has experience but no degree.
  • Has neither experience nor a degree.

Numerical Example

• Let's assume the following:
  • 20 people have a degree and experience.
  • 10 people have a degree but no experience.
  • 15 people have experience but no degree.
  • 30 people have neither experience nor a degree.

Totals

• Total with degrees: $20 + 10 = 30$
• Total without degrees: $15 + 30 = 45$
• Total with experience: $20 + 15 = 35$
• Total with no experience: $10 + 30 = 40$
• Grand total: $30 + 45 = 75$ or $35 + 40 = 75$

Conditional Probability Calculation

• Problem: What is the probability that someone has a degree given that they have experience?
• Notation: $P(\text{Degree} | \text{Experience})$
• Using the formula: $P(\text{Degree} | \text{Experience}) = \frac{P(\text{Degree and Experience})}{P(\text{Experience})}$

Calculation Steps

1. Find $P(\text{Degree and Experience})$: 20 out of 75, or $\frac{20}{75}$.
2. Find $P(\text{Experience})$: 35 out of 75, or $\frac{35}{75}$.
3. Calculate the conditional probability: $\frac{\frac{20}{75}}{\frac{35}{75}}$.
4. Simplify: $\frac{20}{75} \div \frac{35}{75} = \frac{20}{75} \times \frac{75}{35} = \frac{20}{35}$.
5. Reduce the fraction: $\frac{20}{35} = \frac{4}{7}$.

Result

• The probability that a person has a degree given that they have experience is $\frac{4}{7}$.

Key Takeaway

• When A has already happened, A becomes the new total sample space (denominator). The conditional probability is the ratio of the intersection of A and B to the probability of A.