Probability of Happening at Least One Independent Event

Data Science and A.I. Lecture Series

By: Bindeshwar Singh Kushwaha

Institute: PostNetwork Academy

1. Probability of Happening at Least One Independent Event

• If \( A \) and \( B \) are independent events, the probability of happening at least one of the events is:
  \[
  P(A \cup B) = 1 – P((A \cup B)^c)
  \]
• Using De Morgan’s law:
  \[
  P(A \cup B) = 1 – P(A^c \cap B^c)
  \]
• Since \( A \) and \( B \) are independent, \( A^c \) and \( B^c \) are also independent:
  \[
  P(A^c \cap B^c) = P(A^c) \cdot P(B^c)
  \]
• Therefore:
  \[
  P(A \cup B) = 1 – \big(P(A^c) \cdot P(B^c)\big)
  \]
• For \( n \) independent events \( A_1, A_2, \dots, A_n \):
  \[
  P(A_1 \cup A_2 \cup \dots \cup A_n) = 1 – \prod_{i=1}^n P(A_i^c)
  \]
• This is equivalent to \( 1 – \) probability of none of the events occurring.

2. Example: Probability the Target is Hit

Problem:

• A person is known to hit the target in 4 out of 5 shots.
• Another person is known to hit the target in 2 out of 3 shots.
• Find the probability that the target is hit when both try.

Solution:

• Let \( A \): First person hits the target. \( P(A) = \frac{4}{5} \).
• Let \( B \): Second person hits the target. \( P(B) = \frac{2}{3} \).
• The probability of at least one hitting the target is:
  \[
  P(A \cup B) = 1 – P(A^c \cap B^c)
  \]
• Compute \( P(A^c) \) and \( P(B^c) \):
  \[
  P(A^c) = 1 – \frac{4}{5} = \frac{1}{5}, \quad P(B^c) = 1 – \frac{2}{3} = \frac{1}{3}
  \]
• Calculate \( P(A^c \cap B^c) \):
  \[
  P(A^c \cap B^c) = P(A^c) \cdot P(B^c) = \frac{1}{5} \cdot \frac{1}{3} = \frac{1}{15}
  \]
• Substitute into the formula:
  \[
  P(A \cup B) = 1 – \frac{1}{15} = \frac{14}{15}
  \]
• The probability that the target is hit is \( \frac{14}{15} \).

PDF

Probability of Happening at Least One Independent Event

Video

3. Reach PostNetwork Academy

• Website: www.postnetwork.co
• YouTube Channel: www.youtube.com/@postnetworkacademy
• Facebook Page: www.facebook.com/postnetworkacademy
• LinkedIn Page: www.linkedin.com/company/postnetworkacademy

Thank You!