Independent vs Dependent Events
Section: Probability | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What are Independent and Dependent Events?
- Independent events: the outcome of one event does NOT affect the probability of the other.
- Dependent events: the outcome of one event DOES affect the probability of the other.
Identifying Independent Events
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Question: A coin is flipped, then a die is rolled. Are these events independent? Explain.
- Answer: Yes, independent - the coin's result (heads/tails) has no effect on which number the die shows
Identifying Dependent Events
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Question: A bag has 5 red and 3 blue balls. One ball is drawn and NOT replaced, then a second ball is drawn. Are these events independent? Explain.
- Answer: No, dependent - after the first draw, there's one fewer ball in the bag, and the ratio of remaining colours changes, affecting the second draw's probabilities
Testing for Independence Using Probabilities
Instead of just guessing, you can PROVE two events are independent by comparing a probability worked out two different ways: once assuming one event has happened, and once assuming it hasn't. If the probability is the SAME both times, the events are independent.
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Question: A ticket numbered 1 to 10 is drawn at random. X: the number is a multiple of 5. Y: the number is even. Show that X and Y are independent.
- Step 1: Suppose X happens: the number is 5 or 10 (2 numbers). Of these, only 10 is even, so P(Y given X) = 1/2
- Step 2: Suppose X does NOT happen: the number is one of 1,2,3,4,6,7,8,9 (8 numbers). Of these, 2,4,6,8 are even (4 numbers), so P(Y given not X) = 4/8 = 1/2
- Step 3: Both give 1/2 - whether X happens or not, P(Y) stays the same
- Answer: X and Y are independent
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Question: Using the same tickets, Z: the number is a multiple of 3. Show that X and Z are NOT independent.
- Step 1: Suppose X happens: the number is 5 or 10. Neither is a multiple of 3, so P(Z given X) = 0/2 = 0
- Step 2: Suppose X does NOT happen: the number is one of 1,2,3,4,6,7,8,9. Of these, 3,6,9 are multiples of 3 (3 numbers), so P(Z given not X) = 3/8
- Step 3: 0 is not equal to 3/8 - P(Z) changes depending on whether X happens
- Answer: X and Z are NOT independent
The Multiplication Rule for Independent Events
For independent events A and B: P(A and B) = P(A) × P(B).
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Question: A coin is flipped and a die is rolled. Find P(heads AND rolling a 6).
- Step 1: P(heads) = 1/2, P(6) = 1/6
- Step 2: P(heads and 6) = 1/2 × 1/6
- Answer: 1/12
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Question: A spinner has 4 equal sections: red, blue, green, yellow. It is spun twice. Find P(red on the first spin AND blue on the second).
- Step 1: P(red) = 1/4, P(blue) = 1/4
- Step 2: P(red then blue) = 1/4 × 1/4
- Answer: 1/16
Successive Events: With and Without Replacement
With replacement: the item is put back before the next draw - probabilities stay the same (independent). Without replacement: the item is not put back - probabilities change (dependent).
Replacing an item keeps probabilities the same; not replacing it changes them
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Question: A bag has 4 red and 6 blue balls (10 total). A ball is drawn and replaced, then a second ball is drawn. Find P(red on the second draw).
- Step 1: Since the ball was replaced, the bag is unchanged: 4 red, 10 total
- Answer: P(red) = 4/10 = 2/5 (same as the first draw - independent)
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Question: Using the same bag, a red ball is drawn and NOT replaced. Find P(red on the second draw).
- Step 1: After removing 1 red ball: 3 red, 6 blue, 9 total
- Answer: P(red) = 3/9 = 1/3 (changed from 4/10 - dependent)
Real-World Applications
Independence and dependence appear throughout probability-based decisions:
- Card games: probabilities of drawing cards without replacement.
- Lottery draws: numbers drawn without replacement.
- Manufacturing: testing products from a batch without replacement.
- Weather: independent daily events vs dependent streak patterns.
- Genetics: independent inheritance of some traits vs linked traits.
Exam Tips
Common Mistakes
MistakeAssuming events are always independent, even without replacement
Fixcheck whether an item is replaced before the next draw - if not, the events are usually dependent
MistakeUsing the same probability for both draws in a "without replacement" scenario
Fixrecalculate the second draw's probability using the updated numbers after the first item is removed
MistakeMultiplying dependent-event probabilities using the ORIGINAL, unchanged probability for both
Fixfor dependent events, the second event's probability must reflect the new totals
MistakeConfusing "independent" with "unrelated in real life" rather than the precise mathematical meaning
Fixindependence specifically means one outcome doesn't change the probability of the other
MistakeForgetting to reduce both the specific count AND the total when removing an item without replacement
Fixafter removing an item, reduce both the relevant count and the overall total
MistakeJudging independence "by feel" instead of comparing probabilities when asked to show or prove it
Fixwork out the probability assuming the first event happens, then assuming it doesn't - if the two match, they're independent
For Exams
- Ask: does the first event change the probabilities for the second? If yes, dependent; if no, independent.
- "With replacement" usually means independent; "without replacement" usually means dependent.
- For independent events: P(A and B) = P(A) × P(B), using the SAME probabilities each time.
- For dependent events: recalculate the second event's probability using updated totals.
- Explain your reasoning clearly when asked to identify independent vs dependent events.
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