Suppose I have 1 Blue and 1 Red die, which I put into the following events:
- Event A: {Blue Die rolled 1 or 2 or 3}
- Event B: {Red Die rolled 1 or 2 or 3}
- Event C: {Red and blue die are equal, and both show 1, 2 or 3}
I know that the $P(A)=\frac { 3 }{ 6 } =\frac { 1 }{ 2 } $ and $P(B)=\frac { 3 }{ 6 } =\frac { 1 }{ 2 } $.
I wish to calculate $P(C)$ intuitively, without relying on the formula $P(C)=P(A)\cdot P(B)$.
I began my attempt to think of the problem 'intuitively' this way:
There are a total of $36$ outcomes for the 2 dices. The probability to get both red and blue dice rolled with value $1$, the probability is only $\frac { 1 }{ 36 } $. So, if I want both red and blue dice to be $1$ or $2$ or $3$, it seems like there is only 3 possibilities out of the 36 combinations of outcomes from the 2 dices. So I use the addition rule: $Pr(C) = \frac { 1 }{ 36 } +\frac { 1 }{ 36 } +\frac { 1 }{ 36 } =\frac { 3 }{ 36 } =\frac { 1 }{ 12 } $.
But if I use the probability formula, event $C$ needs both Red and Blue dices to be 1 or 2 or 3; then $P(C)=P(A\cap B)=\frac { 1 }{ 2 } \times \frac { 1 }{ 2 } =\frac { 1 }{ 4 } $, which is not $\frac { 1 }{ 12 }$!
What is wrong with my 'intuitive' solution?