Question

The probability that in the toss of two dice, we obtain the sum $7$ or $11$ is
1. $\dfrac{1}{6}$
2. $\dfrac{1}{{18}}$
3. $\dfrac{2}{9}$
4. $\dfrac{{23}}{{108}}$

Answer 1

Here we are asked to find the probability of getting the sum seven or eleven when two dice are rolled. In order to get that we will have to find the number of favorable outcomes and the total number of outcomes according to the given experiment and event. Find the probability using the appropriate formula and values. Remember that probability can never be negative.

Complete step-by-step solution:
Sample Space: The sample space associated with a random experiment is the set of all possible outcomes. An event is a subset of the sample space.
Event: An event E is said to occur on a particular trial of the experiment if the outcome observed is an element of the set E.
We know that Probability (event) $= \dfrac{{{\text{Number}}\,{\text{of}}\,{\text{favourable}}\,{\text{outcomes}}}}{{{\text{Total}}\,{\text{number}}\,{\text{of}}\,{\text{outcomes}}}}$
We are to find The probability that in the toss of two dice, we obtain the sum $7$ or $11$ .
Now the first dice may appear in $6$ different ways and according to any one way in which the first appears, the second can appear in $6$ ways.
The two dice therefore may appear in $6 \times 6 = 36$ ways.
Favourable ways of getting a sum of $7$ are $(1,6),(6,1),(2,5),(5,2),(3,4),(4,3)$
Thus the sum of $7$ may appear in $6$ different ways.
P(getting sum of $7$) $= \dfrac{6}{{36}} = \dfrac{1}{6}$
Favourable ways of getting a sum of $11$ are $(5,6),(6,5)$
Thus the sum of $11$ may appear in $2$ different ways.
P(getting sum of $11$) $= \dfrac{2}{{36}} = \dfrac{1}{{18}}$
Hence Probability of getting sum of $7$ or $11$) $= \dfrac{1}{6} + \dfrac{1}{{18}}$
$= \dfrac{2}{9}$
Therefore option (3) is the correct answer .

Note: Multiplication rule of counting is followed when we calculate the total number of outcomes when two dice are rolled. Multiplication rule of counting states that if there are x ways of doing one thing and y ways of doing another thing, then both the things can be done in xy ways. An Additional rule is also followed when we add the probabilities of sum of seven and eleven appearing on the dice. The additional rule of probability states that the probability of either of the two events to happen is the sum of the probabilities of the two events provided that the two events are mutually exclusive.