A binary source transmits either a symbol "1" or "0" at discrete time intervals. The source could transmit independently either symbol with equal probability. What is the probability of transmitting exactly two "1"s in the first five transmitted symbols?

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Solution 1

This is a binomial probability problem. The binomial distribution model is appropriate for a sequence of n

Solution 2

This is a binomial probability problem. The binomial distribution model deals with finding the probability of success of an event which has only two possible outcomes in a series of experiments, such as a coin toss (success/failure, win/lose, yes/no).

In this case, we want to find the probability of exactly 2 successes (transmitting "1") in 5 trials (transmitted symbols).

The formula for binomial probability is:

P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))

where:

P(X=k) is the probability of k successes in n trials
C(n, k) is the number of combinations of n items taken k at a time
p is the probability of success on a single trial
n is the number of trials
k is the number of successes

Given that the source could transmit independently either symbol with equal probability, p = 0.5. n = 5 (the number of transmitted symbols), and k = 2 (the number of "1"s).

Substituting these values into the formula gives:

P(X=2) = C(5, 2) * (0.5^2) * ((1-0.5)^(5-2)) = 10 * 0.25 * 0.125 = 0.3125

So, the probability of transmitting exactly two "1"s in the first five transmitted symbols is 0.3125 or 31.25%.

This problem has been solved

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