Independent Events Math Example 5

Follow the full solution, then compare it with the other examples linked below.

Example 5

hard
Events AA and BB satisfy P(A)=0.6P(A) = 0.6, P(B)=0.7P(B) = 0.7, and P(A∩B)=0.42P(A \cap B) = 0.42. Are AA and BB independent? Justify algebraically.

Solution

  1. 1
    For independence, we need P(A∩B)=P(A)⋅P(B)P(A \cap B) = P(A) \cdot P(B)
  2. 2
    Calculate: P(A)⋅P(B)=0.6×0.7=0.42P(A) \cdot P(B) = 0.6 \times 0.7 = 0.42
  3. 3
    Compare: P(A∩B)=0.42=P(A)⋅P(B)P(A \cap B) = 0.42 = P(A) \cdot P(B), so yes, the events are independent

Answer

Yes, AA and BB are independent since P(A∩B)=P(A)⋅P(B)=0.42P(A \cap B) = P(A) \cdot P(B) = 0.42.
Independence is verified algebraically by checking whether the product of individual probabilities equals the joint probability. This is the formal definition, not just intuition about physical separation.

About Independent Events

Two events are independent if the occurrence of one does not change the probability of the other: P(A∩B)=P(A)⋅P(B)P(A \cap B) = P(A) \cdot P(B).

Learn more about Independent Events →

More Independent Events Examples

Example 1 easy

A fair coin is flipped and a fair die is rolled. Find [formula].

Example 2 medium

For two events [formula] and [formula], [formula] and [formula]. If [formula] and [formula] are inde

Example 3 medium

A coin is flipped and a die is rolled. Find P(heads AND rolling a 6).

Example 4 easy

A student randomly guesses on two multiple-choice questions, each with 4 options. Find the probabili

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