There are two dice (A ) and (B ). Die (A ) has 4 red and 2 white faces and (B ) has 2 red and 4 white faces. A coin is tossed once, if it shows head, die (A ) is rolled, if it shows tail, die (B ) is rolled, if the probability that die (A ) is used is ( ( 32 33 ) ) when it is given that red turns up every time in first (n ) throws, then (n= )

Step-by-step solution

Let events, (E₁ ) : die (A ) is used when head is appeared (E₂: ) die (B ) is used when tail is appeared (R ) : red face appears. ( array llll & (P (E₁ R ) . = 32 33 & P (E₁ ) = 1 2 , P (E₂ )= 1 2 , & P (R E₁ ) = 2 3 2 3 2 3 n times = ( 2 3 )^n array ) Similarly, (P (R E₂ )= ( 1 3 )^n ) ( ) By Baye's theorem ( aligned P (E₁ R ) & = P (E₁ ) P (R E₁ ) P (E₁ ) P (R E₁ )+P (E₂ ) P (R E₂ ) 32 33 & = 2^n 1+2^n n=5 aligned )