Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be 1/2 and probability of occurrence of 0 at the odd place be 1/3. Then the probability that '10' is followed by '01' is equal to :

Solution

Maths sir · 3 steps

The pattern is the block 1001, and it can start on an odd place or an even place, so add both cases.

1. Read the event. '10' followed by '01' means the four digits 1, 0, 0, 1 appear one after another. Each digit's chance depends only on whether its place is odd or even.
2. Digit chances. At an odd place, P(0) = 1/3 and P(1) = 2/3. At an even place, P(0) = 1/2 and P(1) = 1/2.

   Block starts at	Places of 1, 0, 0, 1	Probability
   odd place	odd, even, odd, even	2/3 × 1/2 × 1/3 × 1/2 = 1/18
   even place	even, odd, even, odd	1/2 × 1/3 × 1/2 × 2/3 = 1/18

3. So. The two cases cannot both hold for the same block, so add them: 1/18 + 1/18 = 1/9. The answer is (c).

The idea to keep. When a digit's probability depends on its position, list every way the pattern can sit on the places, multiply within each case, then add the cases.

Answer from the NTA final answer key. Solution by PrepWiser.