Bernoulli Experiments
An important kind of experiment is called a Bernoulli experiment or a Bernoulli trial.
Definition. Bernoulli Experiment/Trial - A probability experiment with only two possible
elementary 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 transmi
• 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” (S) and the other a “failure” (F)
• Notation: P ( success ) = p, P ( failure ) = q, and p + q = 1.
• Because we associate a success with a 1 and a failure with a 0, we can take the sample
space for a Bernoulli trial to be Ω = {0, 1}.
Definition. 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
|Ωk |
=
=
=
=
=
=
{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.