Question

A die thrown three times, the probability of getting a larger number than the previous number each time, is?
A) $\dfrac{5}{54}$
B) $\dfrac{5}{108}$
C) $\dfrac{13}{216}$
D) None of these

Answer 1

Hint: The above question is of probability. Actually probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.
Probability of event to happen P(E) = number of favourable outcomes/total number of outcomes

Complete step by step solution:
The given question is that when a die is thrown three times, then the probability of getting larger number than the previous one each time is?
Now the total number of possible outcomes when a die is thrown three times is:
$\Rightarrow {{6}^{3}}=216$
Here we want our outcomes like that each outcome is greater than the previous one, so the possible outcomes in which each number is greater than previous one are: $\begin{align}
& \Rightarrow \left\{ 1,2,3 \right\},\left\{ 2,3,4 \right\},\left\{ 4,5,6 \right\},\left\{ 3,4,5 \right\},\left\{ 1,2,4 \right\},\left\{ 1,2,5 \right\},\left\{ 1,2,6 \right\},\left\{ 1,3,4 \right\}, \\
 & \left\{ 1,3,5 \right\},\left\{ 1,3,6 \right\},\left\{ 1,4,5 \right\},\left\{ 1,4,6 \right\},\left\{ 1,5,6 \right\},\left\{ 2,3,5 \right\},\left\{ 2,3,6 \right\},\left\{ 2,4,5 \right\},\left\{ 2,4,6 \right\}, \\
 & \left\{ 2,5,6 \right\},\left\{ 3,4,6 \right\},\left\{ 3,5,6 \right\} \\
\end{align}$
So, the number of possible outcomes in which each number is greater than the previous one is $20$.
Now we know the formula to find the probability of an event. Let A be the event in which we are throwing die three times, then the probability of A is:
$\Rightarrow $P (A) = number of possible outcomes/total number of outcomes.
The number of possible outcomes is $20$ and total number outcomes is $216$ , so the probability of A is:
$\begin{align}
  & \Rightarrow P\left( A \right)=\dfrac{20}{216} \\
 & \Rightarrow P\left( A \right)=\dfrac{5}{54} \\
\end{align}$

So, the correct answer is “Option A”.

Note: In these types of questions students made mistakes in the favorable outcomes with total outcomes. And the most important thing which we have to find from the given question otherwise with this we cannot find out the probability of any event. The probability of all the events in a sample space adds up to one. We use probability concepts because many events cannot be predicted with total certainty.