Probability
Independent events, mutually exclusive events
nta_pyq_2023_jan
Grade 12
Question:
Two dice are thrown independently. Let A be the event that the number appeared on the 1st die is less than the number appeared on the 2nd die, B be the event that the number appeared on the 1st die is even and that on the second die is odd, and C be the event that the number appeared on the 1st die is odd and that on the 2nd die is even. Then
the number of favourable cases of the event $(A \cup B) \cap C$ is 6
A and B are mutually exclusive
The number of favourable cases of the events A, B and C are 15, 6 and 6 respectively
B and C are independent
Step-by-Step Solution
Key Concept: Count favorable outcomes for each event; check mutual exclusivity (A$\cap$B could be non-empty) and independence (P(B$\cap$C) vs P(B)P(C))
$n(A)=15$, $n(B)=9$ (not 6; B: 3 even $\times$ 3 odd = 9), $n(C)=9$. Checking option (1): $(A\cup B)\cap C$: $= (A\cap C)\cup(B\cap C)$. $B\cap C = \emptyset$ (B: 1st even, C: 1st odd). $A\cap C$: 1st odd < 2nd even: $(1,2),(1,4),(1,6),(3,4),(3,6),(5,6) = 6$. So favorable cases $= 6$. Answer: (1)
Correct Answer: The number of favourable cases of the event $(A \cup B) \cap C$ is 6