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KVPY2015MathematicsProbability

There are 6 boxes labeled B ₁, ~B ₂, . B ₆ . In each trial, two fair dice D ₁, D ₂ are known. If D₁ shows j and D₂ shows k , then j balls are put into the box B_ k . After n trials, what is the probability that B ₁ contains at most one ball?

Options

  1. A( 5^ n-1 6^ n-1 )+ ( 5^ n 6^ n ) ( 1 6 )
  2. B( 5^ n 6^ n )+ ( 5^ n-1 6^ n-1 ) ( 1 6 )
  3. C( 5^ n 6^ n )+n ( 5^ n-1 6^ n-1 ) ( 1 6 )
  4. D( 5^ n 6^ n )+n ( 5^ n-1 6^ n-1 ) ( 1 6² )

Correct answer

D. ( 5^ n 6^ n )+n ( 5^ n-1 6^ n-1 ) ( 1 6² )

Step-by-step solution

Let P ( B ₁ ) be the probability that B ₁ contains atmost one ball P (B₁ )=P (B₁ . contains 0 balls )+P (B₁ . contains 1 balls) = P ( D ₂ . never show 1 )+ P ( D ₂ . shows 1 once when D ₁ show 1 ) = ( 5 6 )^ n + " C₁ ( 5 6 )^ n-1 1 6 1 6 D

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