Binomial Trial

1. Repeated Trials
Probability
(BINOMIAL Trial)
Engr. Maria Romina P. Angustia

2. Bernoulli Trial
• Or binomial trial is a
random experiment with exactly two
possible outcomes, "success" and
"failure", in which the probability of
success is the same every time the
experiment is conducted.
• It is named after Jacob Bernoulli, a Swiss
mathematician of the 17th century.

3. Examples of Bernoulli trials:
• Flipping a coin. In this context, obverse ("heads")
conventionally denotes success and reverse
("tails") denotes failure. A fair coin has the
probability of success 0.5 by definition. In this
case there are exactly two outcomes.
• Rolling a die, where a six is "success" and
everything else a "failure". In this case there are
six outcomes, and the event is a six; the
complementary event "not a six" corresponds to
the other five outcomes.
• In conducting a political opinion poll, choosing a
voter at random to ascertain whether that voter
will vote "yes"

4. P = 𝑛𝐶𝑟 (𝑝) 𝑟
(𝑞) 𝑛−𝑟
• The probability that
an event will occur
exactly r times out of
n trials.
• Where:
p – probability of
success

5. Example 1: Coin
A Fair coin is
tossed 5
times. What is
the
probability of

6. Solution:
• Given:
p -
1
2
𝑠𝑢𝑐𝑐𝑒𝑠𝑠 q -
1
2
𝑓𝑎𝑖𝑙𝑢𝑟𝑒
n- 5 tosses/trials r- 3 heads(outcome)
• Substitute values to the formula:
P(3h) = 5𝐶3 (0.5)3
(0.5)5−3
P(3h) =
𝟓
𝟏𝟔
or 0.3125 or 31.25% (Ans.)

7. Example 2: Dice
Suppose you toss
a toss a pair of
dice 6 times.
What is the
probability that

8. Solution:
• Given:
p -
1
6
𝑠𝑢𝑐𝑐𝑒𝑠𝑠 q -
5
6
𝑓𝑎𝑖𝑙𝑢𝑟𝑒
n- 6 tosses/trials r- 3 times(outcome)
• Substitute values to the formula:
P = 6𝐶3 (
1
6
)3
(
5
6
)6−3
P =
𝟔
𝟏𝟏𝟔𝟔𝟒
𝒐𝒓 . 𝟎𝟓𝟑𝟔 𝒐𝒓 𝟓. 𝟑𝟔% (Ans.)

9. Example 3: Multiple Choice Test
You are taking a 10
question multiple
choice test. If
each question has
four choices and
you guess on each

10. Solution:
• Given:
p -
1
4
𝑠𝑢𝑐𝑐𝑒𝑠𝑠 q -
3
4
𝑓𝑎𝑖𝑙𝑢𝑟𝑒
n- 10 questions r- 7 correct answers
• Substitute values to the formula:
P = 10𝐶7 (
1
4
)7
(
3
4
)10−7
P =
𝟒𝟎𝟓
𝟏𝟑𝟏𝟎𝟕𝟐
𝒐𝒓 . 𝟎𝟎𝟑𝟏 𝒐𝒓 . 𝟑𝟏% (Ans.)