Edexcel IAL - Statistics 1- 3.3 Independent Events- Study notes - New syllabus
Edexcel IAL – Statistics 1- 3.3 Independent Events -Study notes- New syllabus
Edexcel IAL – Statistics 1- 3.3 Independent Events -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 3.3 Independent Events
Independence of Two Events
Two events are said to be independent if the occurrence of one event does not affect the probability of the other.
Definitions and Key Results
Events \( A \) and \( B \) are independent if any (and hence all) of the following equivalent conditions are satisfied:
- \( P(B \mid A) = P(B) \)
- \( P(A \mid B) = P(A) \)
- \( P(A \cap B) = P(A)\,P(B) \)
In practice, independence is most commonly tested using the condition
\( P(A \cap B) = P(A)\,P(B) \)
Important Distinction
Independent events are not the same as mutually exclusive events.
If two events are mutually exclusive and both have non-zero probability, they cannot be independent.
Interpretation
If two events are independent:
- Knowledge that one event has occurred gives no extra information about the other
- The probability of one event remains unchanged
Example :
A fair coin is tossed and a fair six-sided die is rolled. Let \( A \) be the event that the coin shows heads and \( B \) the event that the die shows a 6. State whether \( A \) and \( B \) are independent.
▶️ Answer/Explanation
The result of the coin toss does not affect the result of the die roll.
\( P(A) = \dfrac{1}{2},\quad P(B) = \dfrac{1}{6} \)
\( P(A \cap B) = \dfrac{1}{12} = \dfrac{1}{2} \times \dfrac{1}{6} \)
Conclusion: Events \( A \) and \( B \) are independent.
Example :
The probability that a randomly chosen student studies Biology is 0.5. The probability that a student studies Chemistry is 0.4. The probability that a student studies both subjects is 0.2. Determine whether the events are independent.
▶️ Answer/Explanation
For independence, check whether
\( P(A \cap B) = P(A)\,P(B) \)
\( 0.5 \times 0.4 = 0.2 \)
Conclusion: The events are independent.
Example :
Two events \( A \) and \( B \) satisfy \( P(A) = 0.6 \), \( P(B) = 0.5 \), and \( P(A \cap B) = 0.4 \). Determine whether the events are independent.
▶️ Answer/Explanation
\( P(A)\,P(B) = 0.6 \times 0.5 = 0.3 \)
But \( P(A \cap B) = 0.4 \).
Conclusion: The events are not independent.