Download Class Practice on Binomial Experiments #1

Class Practice on Binomial Experiments #1
Binomial Experiments play an important role in the field of statistics. A binomial
experiment is an experiment where:
1. There are n identical trials of the experiment.
2. Each trial results in one of two possible outcomes (these two outcomes are
called success and failure).
3. The probability of success on any particular trial will be denoted by p. The
probability of failure on any particular trial will be denoted by q. Since there
are only two possible outcomes on any particular trial, p + q will always add
up to 1.
4. The probability of success, p, must remain the same from to trial to trial.
This means that the trials are independent.
5. When the experiment is performed n times, we will be interested in the
number of successes, x, that occur during the n trials.
6. We will also be interested in computing the probability of obtaining x
successes out of the n identical trials. This will be denoted by the
symbol P(x),
⎛n⎞
where P ( x) = ⎜ ⎟ p x q n− x , for x = 0, 1, 2, 3, ... , n.
⎝ x⎠
⎛n⎞
n!
where ⎜ ⎟ =
x !(n − x)!
⎝ x⎠
Example: A coin will be tossed 10 times. Let us assume that “head” and “tail”
are equally likely on any individual toss. However, we are interested in counting
heads.
Trial
n
Success
Failure
p
q
x
Example: It is known that 40% of all college graduates have taken a statistics
class. Ten graduates are selected at random and asked if they have taken a
statistics class.
Trial
n
Success
Failure
p
q
x
Example: For the example above, what is the probability that exactly three of the
ten people polled have taken a statistics class.
Example: For the previous example, determine the probability distribution for
the random variable x = number of college graduates that have taken a statistics
class.
Example: According to a recent study, 70% of all homes in a certain county have
high levels of radon gas leaking into their basements. Four homes are selected
at random and tested for radon.
a. What is the probability that exactly one home will have a high level of radon?
b. What is the probability that at least one home will have a high level of radon?