Assertion (A): Two coins are tossed simultaneously. The probabili… - Vidyadip

MCQ

Assertion (A): Two coins are tossed simultaneously. The probability of getting two heads, if it is known that at least one head comes up, is $\frac{1}{3}$.
Reason $(R)$ : Let $E$ and $F$ be two events with a random experiment, then $P(F / E)=\frac{P(E \cap F)}{P(E)}$.

A
Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
B
Both Assertion (A) and Reason (R) are true, but Reason $(R)$ is not the correct explanation of the Assertion (A).
C
Assertion (A) is true and Reason (R) is False.
D
Assertion (A) is false but Reason (R) is true.

✓

Answer

Sample space $=\{ HH , HT , TH , TT \}$
Let $A$ be the event of coming up two heads
$\therefore \quad A=\{H H\} \Rightarrow P(A)=\frac{1}{4}$
and $B$ be the event of coming up atleast one head
$
\therefore \quad B=\{ HH , HT , TH \} \Rightarrow P(B)=\frac{3}{4}$
Also, $A \cap B=\{H H\} \Rightarrow P(A \cap B)=\frac{1}{4}$
So, required probability $=P(A / B)=\frac{P(A \cap B)}{P(B)}=\frac{\frac{1}{4}}{\frac{3}{4}}=\frac{1}{3}$
So, assertion is true.
Also, reason is true and it is the correct explanation of assertion.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Assertion (A) & Reason (B) MCQ" from Probability→Probability — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: Let $\text{A}^{-1}=\begin{bmatrix}5&-7\\-2&3\end{bmatrix}$ and $\text{B}^{-1}=\begin{bmatrix}7&6\\8&7\end{bmatrix}$ then $(\text{AB})^{-1}=\begin{bmatrix}23&31\\26&35\end{bmatrix}.$
Reason: $(\text{AB})^{-1}=\text{A}^{-1}\text{B}^{-1}.$

Directions : In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices :
Assertion : If $ (\vec{\text{a}}\times\vec{\text{b}})+(\vec{\text{a}}.\vec{\text{b}})=400$ and $|\vec{\text{a}}|=4,$ then $|\vec{\text{b}}|=9.$
Reason : If $\vec{\text{a}}$ and $\vec{\text{b}}$ are any two vectors, then $(\vec{\text{a}}\times\vec{\text{b}})^2$ is equal to $(\vec{\text{a}})^2(\vec{\text{b}})^2-(\vec{\text{a}}.\vec{\text{b}})^2.$

Assertion $(A) : x \sin x \frac{d y}{d x}+(x+x \cos x+\sin x)$ $y=\sin x, y\left(\frac{\pi}{2}\right)=1-\frac{2}{\pi} \Rightarrow y=\frac{x-\sin x}{x(1-\cos x)}$
Reason $(R) :$ The differential equation is linear with integrating factor $x(1-\cos x)$.

Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $A$ relation $R = \{(1, 1), (1, 2), (2, 2), (2, 3), (3, 3)\}$ defined on the set $A = \{1, 2, 3\}$ is symmetri.
Reason: $A$ relation $R$ on the set $A$ is symmetric $(\text{a},\text{b})\in\text{R} \Rightarrow(\text{b},\text{a})\in\text{R}.$

Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A) :$ The function $f(x) = x^3 - 3x^2 + 6x - 100$ is strictly increasing on $R$
Reason $(R) :$ A strictly increasing functions is an injective function.

Assertion (A): Let $A=\{2,4,6\}$ and $B=\{3,5,7,9\}$ and defined a function $f=\{(2,3),(4,5),(6,7)\}$ from $A$ to B. Then, f is not onto.
Reason (R): A function $f$ : $A \rightarrow B$ is said to be onto, if every element of $B$ is the image of some elements of $A$ under $f$.

Directions : In the following questions, the Assertions $(A)$ and Reason $(s)\ (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following :
Assertion $(A) : $ if $\text{y}=\tan^{-1}\Big(\frac{\cos\text{x}+\sin\text{x}}{\sin\text{x}-\cos\text{x}}\Big) ,\frac{-\pi}{4} < \text{x} < \frac{\pi}{4},\text{then}\frac{\text{dy}}{\text{dx}}=-1$
Reason $(R) : \frac{\cos\text{x}+\sin\text{x}}{\sin\text{x}-\cos\text{x}}=\tan\Big(\text{x}+\frac{\pi}{4}\Big)$

Assertion (A): A function $f : N \rightarrow N$ be defined by $f(n)=\left\{\begin{array}{ll}\frac{n}{2} & \text { if } n \text { is even } \\ \frac{(n+1)}{2} & \text { if } n \text { is odd }\end{array}\right.$ for all $n \in N$; is one-one
Reason (R): A function $f: A \rightarrow B$ is said to be injective if $a \neq b$ then $f(a) \neq f(b)$.

Assertion $(A)$ : If $e^{x y}+\log (x y)+\cos (x y)+5=0$, then $\frac{d y}{d x}=-\frac{y}{x}$.
Reason $ (R) : \frac{d}{d x}(x y)=0 \Rightarrow \frac{d y}{d x}=\frac{-y}{x}$

Assertion (A) : Integrating factor of $\left(1+x^2\right) \frac{d y}{d x}+x y=\frac{1}{2 x}$ is given by $\sqrt{1+x^2}$.
Reason (R) : Integrating factor of $\frac{d y}{d x}+P(x) \cdot y=Q(x)$ is $e^{\int P(x) d x}$.