Question

Question: What is a conditional Probability that a randomly generated bit string of length four contains at least two consecutive 0s, given that the first bit is a 1?(Assume the probabilities of a 0 and a 1 are the same.)

Short answer

Answer

The Required Probability is $\frac{3}{8}$.

Step-by-step solution

Given Information

Length of a bit string is 4.

It contains at least two consecutive 0s.

Definition of a Conditional Probability

A Conditional probability is a measure of the probability of an event occurring, given that another event has already occurred.

Calculating the Probability

Total number of bit strings of Length four is 16.

Let S be the set of all bit strings of length four.

Let A be the event that a randomly generated has 1 in the first position.

Let B be the event that a randomly generated string has two consecutive 0s.

Number of strings starting with 1 in S = 8

$P\left(A\right)=\frac{8}{16}=\frac{1}{2}$

Strings in S starting with 1 and having two consecutive 0s are 1000, 1001, 1100.

$P\left(A\cap B\right)=\frac{3}{16}$

Conditional Probability that randomly generated bit string of length four contains at least two consecutive 0s = $\frac{P\left(A\cap B\right)}{P\left(A\right)}=\frac{\frac{3}{16}}{\frac{1}{2}}=\frac{3}{8}$

Thus, the required Probability is $\frac{P\left(A\cap B\right)}{P\left(A\right)}=\frac{\frac{3}{16}}{\frac{1}{2}}=\frac{3}{8}$.