# Ex.15.1 Q24 Probability Solution - NCERT Maths Class 10

## Question

A die is thrown twice. What is the probability that

(i) \(5\) will not come up either time?

(ii) \(5\) will come up at least once?

[**Hint:** Throwing a die twice and throwing two dice simultaneously are treated as the same experiment]

## Text Solution

**What is known?**

A die is thrown twice.

**What is unknown?**

The probability that

(i) \(5\) will not come up either time?

(ii) \(5\) will come up at least once?

**Reasoning:**

Probability of an event

\[=\frac{\begin{bmatrix} \text { Number of}\\ \text{ possible outcomes }\end{bmatrix} }{ \begin{bmatrix}\text { Total no of} \\ \text{favorable outcomes} \end{bmatrix} }\]

Probability of not happening an event is equal to \(1\) minus probability of happening an event.

**Steps:**

(i) No of possible outcomes when \(5\) will come up either time

\[\begin{align} & = \begin{bmatrix} \left( 5,1 \right),\left( 5,2 \right),\left( 5,3 \right),\left( 5,4 \right), \\ \left( 5,5 \right), \left( 5,6 \right),\left( 1,5 \right),\left( 2,5 \right), \\ \left( 3,5 \right),\left( 4,5 \right),\left( 6,5 \right) \end{bmatrix} \\&=11\end{align}\]

probability that \(5\) will come up either time

\[\begin{align}& =\frac{\begin{bmatrix} \text { Number of}\\ \text{ possible outcomes }\end{bmatrix} }{ \begin{bmatrix}\text { Total no of} \\ \text{favorable outcomes} \end{bmatrix} } \\ & =\frac{11}{36} \end{align}\]

probability that \(5\) will not come up either time

\[\begin{align} & =1-\frac{11}{36} \\{} & =\frac{25}{36} \\\end{align}\]

(ii) No of possible outcomes when \(5\) will come up at-least once = \(11\)

probability that \(5\) will come up at-least once

\[\begin{align}& =\frac{\begin{bmatrix} \text { Number of}\\ \text{ possible outcomes }\end{bmatrix} }{ \begin{bmatrix}\text { Total no of} \\ \text{favorable outcomes} \end{bmatrix} } \\ & =\frac{11}{36} \end{align}\]

probability that \(5\) will not come up either time is \(\begin{align}\frac{25}{36}\end{align}\) and probability that \(5\) will come up at least once is \(\begin{align}\frac{11}{36}\end{align}\)