• Balanced Incomplete Block Design (BIBD) Verification:
• BIBD is defined by 5 parameters: $v$ treatments, $b$ blocks, $k$ plots per block ($k \lt v$), $r$ replications per treatment, and $\lambda$ pairs per block, satisfying:

• Evaluating Treatment Frequencies in Existing Blocks:
• Replications ($r = 3$ each):
• Treatment 1 appears in: Block 1, Block 3 (current count = 2 $\implies$ needed in Block 4)
• Treatment 2 appears in: Block 1, Block 2 (current count = 2 $\implies$ needed in Block 4)
• Treatment 3 appears in: Block 2, Block 3 (current count = 2 $\implies$ needed in Block 4)
• Treatment 4 appears in: Block 1, Block 2, Block 3 (current count = 3 $\implies$ fully replicated, cannot appear in Block 4)
• Pair Concurrence ($\lambda = 2$):
• Pair (1, 4): Blocks 1, 3 (count = 2)
• Pair (2, 4): Blocks 1, 2 (count = 2)
• Pair (3, 4): Blocks 2, 3 (count = 2)
• Pair (1, 2): Block 1 only (count = 1 $\implies$ must co-occur in Block 4)
• Pair (2, 3): Block 2 only (count = 1 $\implies$ must co-occur in Block 4)
• Pair (1, 3): Block 3 only (count = 1 $\implies$ must co-occur in Block 4)
• Therefore, Block 4 must contain treatments {1, 2, 3} (Option 3: 2 3 1).

Correct option: (3) 231

• Expectation of Geometric Distribution:
• Let $X$ denote the number of independent tosses required until the first head appears in tossing a fair coin.
• The probability of getting a head on any single toss is $p = 0.5$, and tails is $q = 1 - p = 0.5$.
• The random variable $X$ follows a Geometric distribution:

• The expected value (mean number of tosses) for a geometric distribution is:

• Conclusion: The expectation of the number of tosses required is 2 (Option 3).

Correct option: (3) 2