Question

Two unbiased dice are thrown simultaneously to get the coordinates of a point on the xy-plane. Then the probability that this point lies inside or on the region bounded by \[|x| + |y| = 3 \]

Answer 1

Hint: First write the total number of outcomes that will form after throwing two unbiased dice. We get 36 outcomes. Given \[|x| + |y| = 3 \] , meaning that \[x + y \] will lie in between -3 and +3 only. Since we don’t take negative values in probability, the only possible values are 1, and 2 only. Find the probability of its occurrence which will be P (E), the point lies inside or on the given region.

Complete step-by-step answer:
It is said that two unbiased dice are thrown. The total number of possible outcome from one dice is
6.
Therefore, the possible outcome that can be obtained from two dice is \[6 \times 6 = 36 \] .
The outcomes are listed below
$[(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6) \\
  (3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6), \\
  (5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)]$
P (E), the point lies inside or on the given region, which we need to find.
We know, \[{ \text{Probability P(E) = }} \dfrac{{{ \text{(Number of favourable outcomes)}}}}{{({ \text{Total number of possible outcomes}})}} \]
Total number of possible outcomes=36
To find favourable outcome,
In above listed outcomes we can see that the only possible cases are \[(1,1) \] , \[(1,2) \] and \[(2,1) \] .
Because from the given condition \[|x| + |y| = 3 \] we can only take 1 and 2.
Hence, total number of favourable outcomes =3
Now, substituting in the above formula, we get
 \[P(E) = \dfrac{3}{{36}} \]
 \[ \Rightarrow P(E) = \dfrac{1}{{12}} \] Is the required answer.
Note: They give the same type of questions but with different conditions. In this problem we have probability that this point lies inside or on the region bounded by \[|x| + |y| = 3 \] . Focus on the condition which they have given. So that only the number of favourable outcomes will change and rest is done by the same procedure that mentioned above.
So, the correct answer is “$ \dfrac{1}{{12}}$”.

Note: They give the same type of questions but with different conditions. In this problem we have probability that this point lies inside or on the region bounded by \[|x| + |y| = 3 \] . Focus on the condition which they have given. So that only the number of favourable outcomes will change and rest is done by the same procedure that mentioned above.