Binominal Distributions Worksheet With Answer Key - Helm 2008 Section 372 Page 2
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201. The binomial model
We have introduced random variables from a general perspective and have seen that there are two
basic types: discrete and continuous. We examine four particular examples of distributions for
random variables which occur often in practice and have been given special names. They are the
binomial distribution, the Poisson distribution, the Hypergeometric distribution and the Normal
distribution. The first three are distributions for discrete random variables and the fourth is for a
continuous random variable. In this Section we focus attention on the binomial distribution.
The binomial distribution can be used in situations in which a given experiment (often referred to,
in this context, as a trial) is repeated a number of times. For the binomial model to be applied the
following four criteria must be satisfied:
the trial is carried out a fixed number of times n
the outcomes of each trial can be classified into two ‘types’ conventionally named ‘success’ or
‘failure’
the probability p of success remains constant for each trial
the individual trials are independent of each other.
For example, if we consider throwing a coin 7 times what is the probability that exactly 4 Heads
occur? This problem can be modelled by the binomial distribution since the four basic criteria are
assumed satisfied as we see.
here the trial is ‘throwing a coin’ which is carried out 7 times
the occurrence of Heads on any given trial (i.e. throw) may be called a ‘success’ and Tails
called a ‘failure’
1
the probability of success is p =
and remains constant for each trial
2
each throw of the coin is independent of the others.
The reader will be able to complete the solution to this example once we have constructed the general
binomial model.
The following two scenarios are typical of those met by engineers. The reader should check that the
criteria stated above are met by each scenario.
1. An electronic product has a total of 30 integrated circuits built into it. The product is capable
of operating successfully only if at least 27 of the circuits operate properly. What is the
probability that the product operates successfully if the probability of any integrated circuit
failing to operate is 0.01?
2. Digital communication is achieved by transmitting information in “bits”. Errors do occur in
data transmissions. Suppose that the number of bits in error is represented by the random
variable X and that the probability of a communication error in a bit is 0.001. If at most 2
errors are present in a 1000 bit transmission, the transmission can be successfully decoded. If
a 1000 bit message is transmitted, find the probability that it can be successfully decoded.
Before developing the general binomial distribution we consider the following examples which, as you
will soon recognise, have the basic characteristics of a binomial distribution.
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HELM (2008):
Workbook 37: Discrete Probability Distributions
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