Question

In the random experiment of tossing two unbiased dice, let E be the event of getting the sum 8 and F be the event of getting even numbers on both the dice. Then,
(I) $P\left( E \right) = \dfrac{7}{{36}}$
(II) $P\left( F \right) = \dfrac{1}{3}$
Which of the following is the correct statement?
A. Both I and II are correct
B. Neither I nor II is correct
C. I is correct, II is incorrect
D. I is incorrect, II is correct

Answer 1

Here the given question is based on the concept of probability. We have to check whether the given probability conditions of getting the sum 8 and getting even numbers on both when two dice thrown at a time is correct or not. For this, first we need to find the total outcomes of two dice thrown at a time then by using the definition of probability and on further simplification we can check the conditions.

Complete step by step answer:
Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e., how likely they are to happen, using it. Probability can range in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event. The probability formula is defined as the probability of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.
$\text{Probability of event to happen}\,P\left( E \right) = \dfrac{\text{Number of favourable outcomes}}{\text{Total Number of outcomes}}$

Consider the given question: The two unbiased dice are thrown simultaneously then we have to find the probability E be the event of getting the sum 8 and F be the event of getting even numbers on both the dice. If the Three dice are thrown simultaneously, total number of outcomes $= {6^2} = 36$.
(I) The possible outcomes to getting the sum 8 is
E = \left\\{ {\left( {4,4} \right)\left( {6,2} \right)\left( {2,6} \right)\left( {5,3} \right)\left( {3,5} \right)} \right\\} = 5
By the definition of probability
$\Rightarrow \,\,P\left( \text{getting a sum 8} \right) = \dfrac{\text{Total possible outcomes to get sum 8}}{\text{Total number of outcomes}}$
$\therefore \,\,P\left( E \right) = \dfrac{5}{{36}}$ ----(1)

(II) The possible outcomes to getting even numbers on both the dice is
F = \left\\{ {\left( {2,2} \right)\left( {4,4} \right)\left( {6,6} \right)\left( {2,4} \right)\left( {4,2} \right)\left( {6,2} \right)\left( {2,6} \right)\left( {4,6} \right)\left( {6,4} \right)} \right\\} = 9
By the definition of probability
$\Rightarrow \,\,P\left( \text{getting a sum even numbers} \right) = \dfrac{\text{Total possible outcomes to get even numbers}}{\text{Total number of outcomes}}$
$\Rightarrow \,\,P\left( F \right) = \dfrac{9}{{36}}$
Divide both numerator and denominator by 9, then
$\therefore \,\,P\left( F \right) = \dfrac{1}{4}$ ----(2)
Hence, both the given statements are wrong.

Therefore, option B is the correct answer.

Note: The probability is a number of possible values. Candidates must know the knowledge of dice, there are six faces in a single dice and its possible outcomes will be 6, then the total outcomes of three dice are thrown simultaneously are $6 \times 6 = {6^3}$. When we imagine the dice and its thrown the solution will be easy to solve.