Question

# The probability of obtaining an even prime number on each die, when a pair of dice is rolledis:(a) 0(b) $\dfrac{1}{3}$(c) $\dfrac{1}{12}$(d) $\dfrac{1}{36}$

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Hint: We know that 2 is an only even prime number, so first we have to find the probability of getting 2 on each die and then multiply them to get the required value of probability.

Here, we are given that a pair of dice is rolled. We have to find the probability of obtaining an even prime number on each die.
First of all, we know that there is just one even prime number and that is 2. All other prime numbers except 2 are odd.
Therefore, here we have to find the probability of getting 2 on each die.
We know that a die has a total of 6 faces and each face has different numbers that are 1, 2, 3, 4, 5, and 6.
Now, when we roll a die we can get any one of these 6 numbers.
Therefore, the total outcomes of rolling a die = 6.
Now, out of 6 outcomes, there would be just 1 outcome when we would get the number 2 on the die.
Therefore, the probability of obtaining the number 2 when a die is rolled is:
\begin{align} & =\dfrac{\text{No}\text{.of Favorable Outcomes}}{\text{No}\text{.of total outcomes}} \\ & =\dfrac{1}{6} \\ \end{align}
Now, when the other die is rolled, the probability of getting 2 would again be $\dfrac{1}{6}$.
Therefore, when this pair of dice is rolled, the probability of obtaining 2 on each die is
$=\left( \text{Probability of getting 2 on one die} \right)\times \left( \text{Probability of getting 2 on the other die} \right)$
$=\dfrac{1}{6}\times \dfrac{1}{6}$
$=\dfrac{1}{36}$
Hence, option (d) is correct.

Note: This question can also be solved as follows:
When two dice are rolled, the total number of outcomes are
$6\times 6=36$
Now, we know that there would be just 1 out of 36 outcomes when we would get 2 on both dice.
Therefore, the probability of getting 2 on each die
\begin{align} & =\dfrac{\text{No}\text{.of Favorable Outcomes}}{\text{No}\text{.of total outcomes}} \\ & =\dfrac{1}{36} \\ \end{align}
which is the required value.