Two events are independent when the first outcome leaves the second probability unchanged, and dependent when it changes it. To test, ask what is left for the second event after the first has happened.
This lesson belongs to the combined-event probability section. It follows drawing sample spaces systematically.
How do I decide if events are dependent?
Check whether the total, or the number of favourable items, changes after the first event. If either changes, the second probability is different and the events are dependent.
Replacement is the usual clue. If an item is put back, the situation resets, so the events are independent.
Worked example: one box of pens, two readings
A box holds 5 blue pens and 3 black pens, which is 8 pens. Two pens are drawn one after another. Find the probability that both are blue.
With replacement (independent). The first pen is returned, so the box is unchanged.
P(blue, blue) = 5/8 × 5/8 = 25/64.
Without replacement (dependent). After one blue pen is removed, 4 blue pens remain among 7.
P(blue, blue) = 5/8 × 4/7 = 20/56 = 5/14.
The box is the same in both cases. Only the phrase “returned” or “not returned” changes the second fraction.
The mistake that keeps the second fraction fixed
A common slip is to write 5/8 × 5/8 for the no-replacement case. The working looks neat, but the second 5/8 assumes the first pen was put back.
Check the denominator. If no pen is returned, the second denominator must be one less than the first. A second denominator of 8 after a pen has been kept is the error.
Check yourself
A bag holds 2 cards marked X and 4 cards marked Y. Two cards are drawn without replacement. Find P(X then Y), and then find P(two Y) with replacement.
Answer
Without replacement, after an X is removed there are 5 cards, of which 4 are Y.
P(X then Y) = 2/6 × 4/5 = 8/30 = 4/15.
With replacement, P(two Y) = 4/6 × 4/6 = 16/36 = 4/9.
What to study next
Put these products onto a diagram in using tree diagrams with and without replacement. Then try the combined-event probability practice set.
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