Question

Given that the two numbers appearing on throwing two dice are different. Find the probability of the event 'the sum of numbers on the dice is 4'.

Answer 1

Hint: We will first find the probability of the event that two numbers appearing on throwing two dice are different. Then, we will find the probability that the sum of numbers on the dice is 4. We will use the formula for conditional probability to obtain the required answer. We can use conditional probability since we have to find the probability of an event in relation to a given event.

Complete step by step answer:
There are total of 36 outcomes of rolling two dice. Let event A be the event where the two numbers appearing on rolling two dice are different. There are 6 outcomes which have the same number on both dice. Therefore, the probability of getting two different numbers on rolling two dice will be
$P\left( A \right)=\dfrac{36-6}{36}=\dfrac{30}{36}$
Now, let B be the event where the sum of the numbers on the two dice is 4. The outcomes in event B are as follows,
$B=\left\{ \left( 1,3 \right),\left( 2,2 \right),\left( 3,1 \right) \right\}$
The probability of event B occurring is given by,
$P\left( B \right)=\dfrac{3}{36}$
Now, the formula for conditional probability is the following,
$P\left( B|A \right)=\dfrac{P\left( A\cap B \right)}{P\left( A \right)}$
where $P\left( B|A \right)$ is the probability of event B given A.
Now, we have $A\cap B=\left\{ \left( 1,3 \right),\left( 3,1 \right) \right\}$. Therefore, the probability of $A\cap B$ is given by
$P\left( A\cap B \right)=\dfrac{2}{36}$
Next, we will substitute the values of the probability of $A\cap B$ and the probability of even A in the formula above. We have the following,
$P\left( B|A \right)=\dfrac{\left( \dfrac{2}{36} \right)}{\left( \dfrac{30}{36} \right)}$
Simplifying the above equation we get,
$P\left( B|A \right)=\dfrac{1}{15}$
Hence, the probability of event B given that event A occurs is $\dfrac{1}{15}$.

Note:
 It is essential to write different events with distinct names to avoid confusion. The formula for conditional probability is tricky. The denominator has to be the probability of the event that is given. There is a possibility to make mistake in selecting the correct event for the denominator.