Question

In a simultaneous throw of a pair of dice, if the probability of getting the sum less than 7 is $\dfrac{a}{b}$. Find a + b.

Answer 1

Hint: In this question, we will use the basic formula to calculate the probability of occurring an event which is given as: $P(E) = \dfrac{{{\text{Total number of favourable outcomes}}}}{{{\text{Total number of possible outcomes }}{\text{.}}}}$. Then we calculate the total possible events when two dice are thrown simultaneously and find the total number of favourable outcomes.

Complete step-by-step answer:
In the question, two dice are thrown simultaneously. First we will calculate the total number of possible events.
In a throw of pair of dice, total number of possible outcomes = $6 \times 6 = 36$ which are as follow:
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 13), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6).
Let E be the event of getting a sum less than 7.
The sum should be less than 7. So, it can be 6, 5, 4, 3 and 2
Number of favourable outcomes = 15 which are as follow:
{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4) (3, 1), (3, 2), (3, 3), (4, 1), (4, 2), (5, 1)}.
Now, we know that the formula for calculating the probability is given as:
$P(E) = \dfrac{{{\text{Total number of favourable outcomes}}}}{{{\text{Total number of possible outcomes }}{\text{.}}}}$
Putting the value in above formula, we get:
$P(E) = \dfrac{{{\text{15}}}}{{{\text{36}}}} = \dfrac{5}{{12}}$.
On comparing the above value with $\dfrac{a}{b}$, we have a = 5 and b = 12.
So, a + b = 5+12=17.

Note: In this type of question, you should know to calculate the total outcomes when ‘n’ dice are thrown simultaneously. The formula for calculating the total number of outcomes when ‘n’ dice are thrown simultaneously is ${(6)^n}$, where ‘n’ is the number of dice. Probability can never be greater than 1.