Step-by-Step Solution

Topic: Independent Events

1. Define Probabilities

Let A, B, and C be the events that the respective players hit the target.

Player Hit Probability (P) Miss Probability (P’) = 1-P
A 4/5 1/5
B 3/4 1/4
C 2/3 1/3

(I) EXACTLY TWO HIT THE TARGET

There are three mutually exclusive cases where exactly two players hit:

  1. Case 1: A hits, B hits, C misses ⇒ P(A) × P(B) × P(C’)
    = (4/5) × (3/4) × (1/3) = 12/60
  2. Case 2: A hits, B misses, C hits ⇒ P(A) × P(B’) × P(C)
    = (4/5) × (1/4) × (2/3) = 8/60
  3. Case 3: A misses, B hits, C hits ⇒ P(A’) × P(B) × P(C)
    = (1/5) × (3/4) × (2/3) = 6/60

Total Probability: Sum of all cases

P(Exactly 2) = 12/60 + 8/60 + 6/60 = 26/60

Probability = 13/30

(II) AT LEAST ONE HITS THE TARGET

Using the complement rule is much faster here:

P(At least one) = 1 – P(None hits)

Step 1: Find P(None hits)

P(None) = P(A’) × P(B’) × P(C’)

P(None) = (1/5) × (1/4) × (1/3) = 1/60

Step 2: Subtract from 1

P(At least one) = 1 – 1/60

Probability = 59/60

⚠️ Silly Mistake Audit: “At Least One”

A common error in “at least one” problems is trying to calculate every positive case (1 hit + 2 hits + 3 hits). While correct, it is long and prone to calculation errors. Always use the Complement Method (1 – None) for speed and accuracy.

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