Q.Two dice are tossed. Find whether the following two events $A$ and $B$ are independent: $A = \{(x, y) : x + y = 11\}$, $B = \{(x, y) : x \neq 5\}$ where $(x, y)$ denotes a typical sample point.

Test: $A,B$ are independent iff $P(A\cap B)=P(A)\,P(B)$. Two dice give $36$ outcomes.

• $A:\ x+y=11\Rightarrow\{(5,6),(6,5)\}$, so $P(A)=\dfrac{2}{36}=\dfrac{1}{18}$.
• $B:\ x\ne 5\Rightarrow 5\times 6=30$ outcomes, so $P(B)=\dfrac{30}{36}=\dfrac{5}{6}$.
• $A\cap B$: sum $11$ and $x\ne 5$. Of $(5,6),(6,5)$, only $(6,5)$ has $x\ne 5$, so $P(A\cap B)=\dfrac{1}{36}$. …

The independence test

Two events are independent exactly when $P(A\cap B)=P(A)\,P(B)$. So we just compute all three probabilities over the $6\times 6=36$ outcomes of two dice.

$P(A)$: sum equals $11$

The only pairs with $x+y=11$ are $(5,6)$ and $(6,5)$, so

$P(B)$: first die is not $5$

The first coordinate $x$ can be any of $\{1,2,3,4,6\}$ (5 choices) and $y$ any of $6$ values, giving $5\times 6=30$ outcomes:

$P(A\cap B)$: sum $11$ and $x\ne 5$

Of the two outcomes in $A$, the pair $(5,6)$ has $x=5$ and is excluded by $B$; only $(6,5)$ survives. So

Compare …

Method: Testing two events for independence

Use this whenever a problem asks "are events $A$ and $B$ independent?" — it is a computation, not a matter of whether the descriptions "sound related".

Steps

Step 1: Compute the individual probabilities

Find $P(A)$ and $P(B)$ over the full sample space (for two dice, $36$ equally likely outcomes).

Step 2: Compute the joint probability directly …

Common Mistakes

Mistake 1: Confusing independent with mutually exclusive

Why it's wrong: mutually exclusive events (empty intersection) with non-zero probability are actually strongly dependent. Correct approach: test the product rule $P(A\cap B)=P(A)P(B)$, not whether they can co-occur.

Mistake 2: Miscounting $A\cap B$ …