A player participates in 3 matches against three teams T₁, T₂ and T₃.The probability of winning a match against teams T₁, T₂ and T₃ are 0.2, 0.3 and 0.9 respectively. If 'wins' can be regarded as independent events, then the probability that he

(A) wins all the 3 matches is 0.054

(B) wins no match is 0.054

(C) wins exactly two matches is 0.348

(D) wins exactly one match is 0.542

Choose the correct answer from the options given below:

The player plays 3 matches with win probabilities:

Team T₁: P(Win) = 0.2, P(Lose) = 0.8

Team T₂: P(Win) = 0.3, P(Lose) = 0.7

Team T₃: P(Win) = 0.9, P(Lose) = 0.1

Since wins are independent events, probabilities are multiplied.

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Checking statement (A): Wins all 3 matches = 0.054

P(Win all) = 0.2 × 0.3 × 0.9

$= 0.054$

Statement (A) is correct.

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Checking statement (B): Wins no match = 0.054

P(Lose all) = 0.8 × 0.7 × 0.1

$= 0.056$

Statement (B) is incorrect.

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Checking statement (C): Wins exactly 2 matches = 0.348

Three possible scenarios:

Win T₁, Win T₂, Lose T₃: $0.2 × 0.3 × 0.1 = 0.006$

Win T₁, Lose T₂, Win T₃: $0.2 × 0.7 × 0.9 = 0.126$

Lose T₁, Win T₂, Win T₃: $0.8 × 0.3 × 0.9 = 0.216$

P(Exactly 2 wins) $= 0.006 + 0.126 + 0.216$

$= 0.348$

Statement (C) is correct.

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Checking statement (D): Wins exactly 1 match = 0.542

Three possible scenarios:

Win T₁, Lose T₂, Lose T₃: $0.2 × 0.7 × 0.1 = 0.014$

Lose T₁, Win T₂, Lose T₃: $0.8 × 0.3 × 0.1 = 0.024$

Lose T₁, Lose T₂, Win T₃: $0.8 × 0.7 × 0.9 = 0.504$

P(Exactly 1 win) $= 0.014 + 0.024 + 0.504$

$= 0.542$

Statement (D) is correct.

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Statements (A), (C) and (D) are correct.