IPA FE Subject A 2026 Question 1

Each row sums to 1 — for a given input bit the output is either 0 or 1, so the two transition probabilities in that input’s row add up to 1.

Row 0 holds the probabilities for input 0: a is the probability that the output is 0 and b the probability that it is 1. Given input 0, the output must be one of the two and cannot be both, so a + b = 1. Row 1 is the same for input 1: c + d = 1.

Example: a channel that flips a 0 with probability 0.1 and a 1 with probability 0.3 has a = 0.9, b = 0.1, c = 0.3, d = 0.7. Both rows sum to 1; the columns (1.2 and 0.8) and the diagonals (1.6 and 0.4) do not.

Answer (b)

Why not others:

• •(a) — the table holds two separate distributions, one per input, so the four values together sum to 2, not 1; a total of 1 would hold only for joint probabilities of input and output
• •(c) — adds down a column: a and c are conditioned on different inputs, so nothing makes them sum to 1 (0.9 + 0.3 = 1.2 above)
• •(d) — pairs the two correct transmissions and the two errors, which belong to different inputs and are independent (0.9 + 0.7 = 1.6 above)

Key rule: In a transition (conditional probability) table, each row is the probability distribution of the output for one input, so every row, not every column, sums to 1.

Basic Theory · AI-written, may contain errors