A text is made up of the characters α, β, γ, δ and σ with the probability…

Answer: 2.15 bits per symbol.

Method (Huffman coding): combine the two smallest probabilities repeatedly.

• Combine δ (0.08) and α (0.12) → 0.20

• Combine γ (0.15) and node 0.20 → 0.35

• Combine σ (0.25) and node 0.35 → 0.60

• Combine β (0.40) and node 0.60 → 1.00 (root)

Resulting codeword lengths (from the final tree):

• β (0.40): length 1

• σ (0.25): length 2

• γ (0.15): length 3

• α (0.12): length 4

• δ (0.08): length 4

Compute the average length:

• 0.40 × 1 = 0.40

• 0.25 × 2 = 0.50

• 0.15 × 3 = 0.45

• 0.12 × 4 = 0.48

• 0.08 × 4 = 0.32

• Total = 0.40 + 0.50 + 0.45 + 0.48 + 0.32 = 2.15

Therefore the optimal average code length is 2.15 bits per symbol.