In a seven-match series between Australia and South Africa, the probability of South Africa...

The probability of South Africa winning any two consecutive matches is calculated by multiplying the probability of winning each individual match: P(two consecutive wins) = (3/4) × (3/4) = 9/16. This represents the likelihood of South Africa winning at least one pair of back-to-back matches in the seven-match series.

Step-by-step Derivation:
Step 1: Identify the probability of South Africa winning a single match: P(win) = 3/4

Step 2: For back-to-back (consecutive) wins, we need two consecutive victories.

Step 3: Assuming independence of matches, the probability of winning two consecutive matches is:
P(back-to-back wins) = P(win on match i) × P(win on match i+1)
P(back-to-back wins) = (3/4) × (3/4) = 9/16

Step 4: Verify: 9/16 ≈ 0.5625, which is reasonable since (3/4)² = 0.5625 ✓

Note: The question asks for the probability of back-to-back matches (any consecutive pair), not the probability of at least one such pair occurring in all seven matches. The straightforward interpretation is the probability of two consecutive wins = 9/16.