# Probability distribution

2. A system uses one of two networks to send binary information, where only one of those networks is used to send all bits of a single block of 5 bits. When network 1 is used the probability of an error in one bit is 0.1 and when network 2 is used the probability of error in one bit is 0.3. The probability of an error in using network 1 to send a block is 0.6. Different blocks may be sent via one of the two different networks based on the same probability.

a. Find the probability of receiving a block of 5 bits with at least two errors.

Hint: {at least 2} = {2 or more} = {more than 1} = {Not 0 or 1} = {2,3,4,5} This is clearly the correct definition of {at least 2} for this problem.

b. If a block of 5 bits was received with at least two errors, what is the probability that network 2 was used?

Hint: All 5 bits go through the SAME network. (Think about tossing the same coin 5 times).

In order to avoid error propagation, assume that the answer to (a) is 0.4 (this is not the correct answer) and proceed to do part (c)

c. Blocks received with at least two errors are retransmitted. Each re-transmission may be sent via either channel. What is average number of times we have RE-transmit a block until it is received with less than 2 errors?

Hint: re-transmission needs to include {X - 1}

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3. Voice and data calls arrive at a switch following a Poisson model, but different arrival rates of 6 per second for data, and 2 per second for voice.

a. Find the total arrival rate of calls to the switch.

b. Find the probability that in an interval of length 0.2 seconds, zero voice calls and exactly one data call arrive at the switch.

c. Find the probability that in 0.5 seconds exactly 4 calls (of any kind) arrive at the switch.

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4. A system is composed of three links as shown. The links fail independently of each other, with failure probabilities show on each link. Find the failure probability of the system.

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2. A system uses one of two networks to send binary information, where only one of those networks is used to send all bits of a single block of 5 bits. When network 1 is used the probability of an error in one bit is 0.1 and when network 2 is used the probability of error in one bit is 0.3. The probability of an error in using network 1 to send a block is 0.6. Different blocks may be sent via one of the two different networks based on the same probability.

Here binomial random variable can be used to compute the probability for each network. The probability density function of binomial random variable with parameter n and p is given by .

The probability for different values of x are given below.

Network 1

X P(X)

0 0.59049

1 0.32805

2 0.07290

3 0.00810

4 0.00045

5 0.00001

Network 2

X P(X)

0 0.16807

1 0.36015

2 0.30870

3 0.13230

4 0.02835

5 0.00243

a. Find the probability of receiving a block of 5 bits with at least two errors.

Hint: {at least 2} = {2 or more} = {more than 1} = {Not 0 or 1} = {2,3,4,5} This is clearly the correct definition of {at least 2} for this problem.

Probability of receiving block of 5 bits with at least ...

#### Solution Summary

Step by step method for computing probability is given in the answer. The probability distribution and intervals are provided.