Verified Solution • 2025–26 Session moderate

QJEEM23Ma2a8c: JEE Main 2023 Mathematics - Class 12 JEEM23Ma2a8c Solution: Two dice are thrown independently. Let be the event that the…

Two dice are thrown independently. Let A\mathrm{A} be the event that the number appeared on the 1st 1^{\text {st }} die is less than the number appeared on the 2nd 2^{\text {nd }} die, B\mathrm{B} be the event that the number appeared on the 1st 1^{\text {st }} die is even and that on the second die is odd, and C\mathrm{C} be the event that the number appeared on the 1st 1^{\text {st }} die is odd and that on the 2nd 2^{\text {nd }} is even. Then :

Multiple Choice Options

  • A A and B are mutually exclusive
  • B the number of favourable cases of the events A, B and C are 15, 6 and 6 respectively
  • C B and C are independent
  • D the number of favourable cases of the event (A∪B)∩C(\mathrm{A\cup B)\cap C} is 6

Step-by-Step Verified Solution

amp;A={(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)}amp;n(A)=15amp;B={(2,1),(2,3),(2,5),(4,1),(4,3),(4,5),(6,1),(6,3),(6,5)}amp;n(B)=9amp;C={(1,2),(1,4),(1,6),(3,2),(3,4),(3,6),(5,2),(5,4),(5,6)}amp;n(C)=9\begin{aligned} & A=\{(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)\} \\\\ & n(A)=15 \\\\ & B=\{(2,1),(2,3),(2,5),(4,1),(4,3),(4,5),(6,1),(6,3),(6,5)\} \\\\ & n(B)=9 \\\\ & C=\{(1,2),(1,4),(1,6),(3,2),(3,4),(3,6),(5,2),(5,4),(5,6)\} \\\\ & n(C)=9\end{aligned}

(4,5)∈A and (4,5)∈B(4,5) \in A \text { and }(4,5) \in B

∴A\therefore A and BB are not exclusive events

amp;n((A∪B)∩C)=n(A∩C)+n(B∩C)−n(A∩B∩C)=amp;3+3−0=amp;6\begin{aligned}& n((A \cup B) \cap C)=n(A \cap C)+n(B \cap C)-n(A \cap B \cap C) \\\\= & 3+3-0 \\\\= & 6\end{aligned}

Option (D) is correct.

amp;n(B)=936,n(C)=936,n(B∩C)=0amp;⇒n(B)⋅n(C)≠n(B∩C)amp;∴B and C are not independent. \begin{aligned}& n(B)=\frac{9}{36}, n(C)=\frac{9}{36}, n(B \cap C)=0 \\\\& \Rightarrow n(B) \cdot n(C) \neq n(B \cap C) \\\\& \therefore B \text { and } C \text { are not independent. }\end{aligned}

Related Learning Resources & Exam Preparation