Let a computer program generate only the digits

Question:

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 $\frac{1}{2}$ and probability of occurrence of 0 at the odd place be $\frac{1}{3}$. Then the probability that ' 10 ' is followed by '01' is equal to :

  1. (1) $\frac{1}{18}$

  2. (2) $\frac{1}{3}$

  3. (3) $\frac{1}{6}$

  4. (4) $\frac{1}{9}$


Correct Option: , 4

Solution:

$\begin{array}{cccc}1 & 0 & 0 & 1 \\ \text { odd place even place } & \text { odd place } & \text { even place }\end{array}$

or

$\begin{array}{ccc}1 & 0 & 0 \\ \text { even place odd place } & \text { even place odd place }\end{array}$

$\Rightarrow\left(\frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{2} \cdot \frac{2}{3}\right)+\left(\frac{2}{2} \cdot \frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{2}\right)$

$\Rightarrow \frac{1}{9}$

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