Bernoulli Experiments
An important kind of experiment is called a Bernoulli experiment
or a Bernoulli trial.
Defn. Bernoulli Experiment/Trial - A probability experiment with
only two possible outcomes.
• Examples of Bernoulli Experiments:
– Toss a coin: Ω = {H, T }
– Sent a message through a network and record whether or not it
is received: Ω = {successful transmission, unsuccessful transmission}
– Draw a part from an assembly line and record whether or not
it is defective: Ω = {defective , good}
• Standard Conventions:
– Label one outcome a “success” and the other a “failure”
– Notation: P ( success ) = p, P ( failure ) = q, and p + q = 1.
– Indicator functions:
(
I(ω) =
1 if ω a success
0 if ω a failure
– Because we associate a success with at 1 and a failure with
a 0, we can take the sample space for a Bernoulli trial to be
Ω = {0, 1}.
Defn. Sequence of Bernoulli Experiments - A compound experiment consisting of n independent and identically distributed
Bernoulli experiments.
• Examples of Sequences of Bernoulli Experiments:
– Toss a coin n times.
– Send 23 identical messages through the network independently.
– Draw 5 cards from a standard deck with replacement and
record whether or not the card is a king.
• Comments
– Saying that the trials are independent means, for example,
that
P ( trial 1 a success and trial 2 a failure , and . . . trial k a failure) =
P ( trial 1 a success)P ( trial 2 a failure ) . . . P ( trial k a failure).
– Saying that the trials are identically distributed means that
P ( trial 1 a success) = P ( trial 2 a success ) = . . .
= P ( trial k a success) = p
P ( trial 1 a failure) = P ( trial 2 a failure ) = . . .
= P ( trial k a failure) = q = 1 − p
– Shorthand for “independent and identically distribuded” is
“iid.”
• Sample Space: Ωk for a sequence of k Bernoulli experiments.
Ω1
|Ω1|
Ω2
|Ω2|
Ω3
|Ω3|
...
Ωk
|Ω |
=
=
=
=
=
=
{0, 1}
2
{00, 01, 10, 11}
4
{000, 001, 010, 100, 110, 101, 011, 111}
8
= { k-digit binary numbers}
= 2k
• Probability Assignments
– For a single Bernoulli trial, P (1) = p, P (0) = q
– For a sequence of two Bernoulli trials,
P (00) = q 2,
P (01) = qp,
P (10) = pq,
P (11) = p2
WHY? Independence!
Because the Bernoulli trials are independent,
P ( trial 1 = x1, and trial 2 = x2) = P ( trial 1 = x1)P ( trial 2 = x2)
– For a sequence of k Bernoulli trials,
P (ω) = pj q k−j ,
where ω is a sequence of 1’s and 0’s, j is the number of 1’s,
and k − j is the number of zeros.