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23. Given,

$P(A)=\frac{1}{2}$

$P(A\cup B)=\frac{3}{5}$

$P(B)=p$

(i) We know that,

when $A$ and $B$ are mutually exclusive

$(A\cap B)=\varnothing$ $,P(A\cap B)=0$

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

$\Rightarrow \frac{3}{5}=\frac{1}{2}+p$

$\Rightarrow p=\frac{3}{5}-\frac{1}{2}=\frac{6-5}{10}=\frac{1}{10}$

(ii) when $A$ and $B$ are independent

$\begin{array}{l}P(A\cap B)=P(A).P(B)\\ P(A\cap B)=\frac{1}{2}\times p\end{array}$

Now,

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

$\Rightarrow \frac{3}{5}=\frac{1}{2}+p-\frac{1}{2}p$

$\Rightarrow \frac{3}{5}-\frac{1}{2}=\frac{2p-p}{2}$

$\Rightarrow \frac{6-5}{10}=\frac{p}{2}$

$\Rightarrow p=\frac{1\times 2}{10}=\frac{2}{10}=\frac{1}{5}$

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Yes, the MCNUJC Bhopal BTech admission is open for the academic year 2025. Interested candidates can visit the official website to apply online for the programme. However, it is essential for candidates to submit the application before the stipulated timeline. Those who are eligible must ensure they present the required documents for verification.

$$22 . $P(F)=\frac{3}{10}$

$P(E\cap F)=\frac{1}{5}$

$P(E).P(F)=\frac{3}{5}\times \frac{3}{10}$

$=\frac{9}{50}\ne P(E\cap F)$

Here,  $P(E).P(F)\ne P(E\cap F)$ . $A$ and $B$ are not independent.

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NOTE: The above data is obtained from the NIRF report 2025

21. When die is thrown, the sample space

$S=\{1,2,3,4,5,6\}$

Let,  $A$ =the number is even $=\{2,4,6\}$

$B$ =the number is red $=\{1,2,3\}$

$\begin{array}{l}P(A)=\frac{3}{6}=\frac{1}{2}\\ P(B)=\frac{3}{6}=\frac{1}{2}\\ A\cap B=\left\{2\right\}\\ P(A\cap B)=\frac{1}{6}\\ \therefore P(A).P(B)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}\ne P(A\cap B)\end{array}$

Hence,  $A$ and $B$ are not independent.

20. The sample space of given condition are

$S=\left\{\begin{array}{l}(H,1),(H,2),(H,3),(H,4),(H,5),(H,6)\\ (T,1),(T,2),(T,3),(T,4),(T,5),(T,6)\end{array}\right\}$

Let $A$ = head appear on the coin

$B$ = $3$ on the die

$A=\{(H,1),(H,2),(H,3),(H,4),(H,5),(H,6)\}$

$P(A)=\frac{6}{12}=\frac{1}{2}$

$B=\{(H,3),(T,3)\}$

$P(B)=\frac{2}{12}=\frac{1}{6}$

$A\cap B=\{(H,3)\}$

$i.e.,P(A\cap B)=\frac{1}{12}$

$P(A).P(B)=\frac{1}{2}\times \frac{1}{6}=\frac{1}{12}=P(A\cap B)$

Therefore, $A$ and $B$ are independent.

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19. Let $A$ ,  $B$ and $C$ be the respective events that the first, second and third drawn orange is good.

$P(A)=$ Probability that first drawn orange is good= $\frac{12}{15}$

The orange is not replaced;

$P(B)=$ Probability of getting second orange is good= $\frac{11}{14}$

Similarity, probability of getting third orange is good,  $P(C)=\frac{10}{13}$

$\therefore$ Probability of getting all orange good= $\frac{12}{15}\times \frac{11}{14}\times \frac{10}{13}$ $=\frac{44}{91}$

Therefore, probability that will approve for sale $=\frac{44}{91}$

Once all the IHM Bangalore BSc seats have been allocated via the NCHMCT JEE counselling process, students must report to the institute for verification of documents. These documents include:

• Class 10 and 12 marksheets
• Transfer/Migration and Character Certificate (original)
• NCHMCT JEE scorecard
• Seat allotment letter
• Category Certificate (if applicable)
• Receipt/proof for the payment made by candidates at the time of seat allotment for adjustment
• Passport size photographs

For B.Sc (Hons.) Agriculture at BHU, candidates usually need a CUET UG score of 630–650 marks (around 96–98 percentile) out of 750 in the general category. On the scaled range, the expected cut-off is 499–526 marks. Overall, aiming for above 85% is considered safe to secure admission in this program.

To gain entry into the PG Diploma for an ABEA course, candidates must meet the eligibility criteria set by the institute. Aditya Birla Education Academy offers various one-year courses. Students can gain admission to this course based on merit. In addition to the selection criteria, students must have completed their graduation (common eligibility). 

For admission in a PG Diploma programme at Aditya Birla Education Academy, it is crucial that the aspirant fulfills the course-specific eligibility criteria. Thus, only the asiprants who have passed graduation in any stream are eligible for admission. In addition, student should have passed the qualifying examination from a recognised board. Candidate must also hold the graduation mark sheet during application form filling and at the time of admission for verification purposes.

If you belong to the General AI category, the NEET SS Round 1 cutoff for the DNB course ranged between 252 and 1098. So, considering the first round cutoff, candidates must target a cutoff rank of up to 1098 to be eligible for DNB admission at Vardhman Mahavir Medical College. Check out the table below to view the specialisation-wise cutoff details for the General category under the AI quota.

18. $P(A)=\frac{3}{5}$

$P(B)=\frac{1}{5}$

As $A$ and $B$ are independent event,

$P(A\cap B)=P(A).P(A)$

$=\frac{3}{5}.\frac{1}{5}$

$=\frac{3}{25}$