Are the events independent?

Test, then conclude with a reason

In 6 of 20 recent exams
Paper 1 or 2
1 to 3 marks

How it is set in the exam

Paper 1 (no calculator) and Paper 2 (calculator) equally (3 exam versions each), 1–3 marks, the last part of a Venn, table or tree question, after the numbers it needs have been found. Common (6 of 20 exam versions). Three times the conclusion was given ("show that … are independent", or "… are not independent"), so the marks were all for the numbers.

Spot it in the exam

Exam phrases to spot (4)

“determine whether $A$ and $B$ are independent”
“show that $A$ and $B$ are independent”
“show that the events are not independent”
“justify your answer”

In disguise (3)

"does the chance of $A$ depend on whether $B$ happens?"
"is $A$ more likely for one group than for the whole population?", with a table of two groups
a two-stage experiment where the question is whether the first stage (which bag) and the result (which colour) are independent

Variants (2)

source: a Venn diagram (probabilities or counts), a two-way table, a two-stage tree, or the formulas
asked: test and conclude, show independent, or show not independent

Looks like it, but isn't (3)

independence is given and there is an unknown to find: Probabilities in terms of an unknown
independence is given, and only a product or a conditional is asked: Independent events, one line
"mutually exclusive": the test is $\mathrm{P}(A \cap B) = 0$ (Venn diagrams)

Worked example

Show independence after the addition rule gives P(B)

Paper 1 (no calculator)

Question

Events $A$ and $B$ are such that $\mathrm{P}(A) = \frac{1}{3}$ and $\mathrm{P}(A \cap B) = \frac{1}{5}$ , and the probability that neither $A$ nor $B$ occurs is $\frac{4}{15}$ .
(a) Find $\mathrm{P}(B)$ . [3]
(b) Show that $A$ and $B$ are independent. [2]

Try it yourself first, then compare.

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Which method?

You already have	Compare
$\mathrm{P}(A)$ , $\mathrm{P}(B)$ and $\mathrm{P}(A \cap B)$ (a Venn diagram, or the formulas)	$\mathrm{P}(A)\,\mathrm{P}(B)$ with $\mathrm{P}(A \cap B)$
a row or column of a table, or a branch of a tree, which is $\mathrm{P}(A \mid B)$	$\mathrm{P}(A \mid B)$ with $\mathrm{P}(A)$

Either comparison scores. Pick the one whose numbers are already on the page.

Steps

Find the numbers
the comparison needs, usually from the parts before.
Write the comparison with the numbers and a sign:
$\mathrm{P}(A)\,\mathrm{P}(B) = 0.1875 \ne 0.225 = \mathrm{P}(A \cap B)$ . Don't round before comparing.
Conclude in words,
with the reason attached: "so $A$ and $B$ are not independent". A conclusion scores only with correct values (markschemes don't allow or ).
Two-stage setting:
a verbal reason is also accepted: the containers have different proportions, so the first stage changes the chance of the result.

Common mistakes

Testing $\mathrm{P}(A) + \mathrm{P}(B) = \mathrm{P}(A \cup B)$
that is the test for mutually exclusive, not independent.
A conclusion with no numbers,
or numbers with no conclusion: both are needed, and the reason scores only when the values are right.
"Independent" and "mutually exclusive" confused.
Mutually exclusive events (with non-zero probabilities) are never independent.

More mistakes (3)

Comparing counts instead of proportions
the same number of green counters in two bags of different sizes is not the same probability.
Rounding before comparing
$0.1875$ and $0.225$ are different, but close values can round to the same thing.
Assuming independence to find $\mathrm{P}(A \cap B)$ , then "testing" it
the test needs $\mathrm{P}(A \cap B)$ from somewhere else.

Easy to confuse with

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Easy to confuse with

More in Probability: Venn diagrams, trees and conditional probability