Page 338 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Unit 26: Theory of Probability: Introduction and Uses

Example 12: What is the probability that a leap year selected at random will contain 53 Sundays ? Notes
Solution: Total number of days in a leap year = 366

366 2
Number of weeks in a year = = 52
7 7
= 52 weeks and 2 days
Following may be the 7 possible combinations of these two extra days:
(i) Monday and Tuesday (ii) Tuesday and Wednesday
(iii) Wednesday and Thursday (iv) Thursday and Friday
(v) Friday and Saturday (vi) Saturday and Sunday
(vii) Sunday and Monday
A selected leap year can have 53 Sundays if these two extra days happen to be a
Sunday
Total possible outcomes of 2 days = n = 7
Number of cases having Sundays = m = 2
2
∴ The required probability =
7
Use of Bernoulli’s Theorem in Theory of Probability

Bernoulli’s theorem is very useful in working out various probability problems. This theorem states
that if the probability of happening of an event in one trial or experiment is known, then the probability
of its happening exactly, 1, 2, 3, ... r times in n trials can be determined by using the formula:
r
n
⋅
P (r)= C pq n −r r = 1, 2, 3, ... n
r
where,
P (r) = Probability of r successes in n trials.
p = Probability of success or happening of an event in one trial.
q = Probability of failure or not happening of the event in one trial.
n = Total number of trials.
The following examples illustrate the applications of this theorem:
Example 13: The chance that a ship safely reaches a port in 1/5. Find the probability that out of 5
ships expected at least one would arrive safely.

1 1 4
1
Solution: Given, n = 5, p = , q = − =
5 5 5
P (at least one ship arriving safely) = 1 – 1 (none arriving safely)
5
0
⋅
= − 1 ⎡ 5 C 0 () ( ) )
p
q
⎢ ⎣
⎡ ⎛⎞ ⎛ ⎞ 5 ⎤ ⎛⎞ 5
0
4
1
4
= − ⎢ 1 5 0 ⎜⎟ ⎜ ⎟ ⎥ C = − ⎜⎟
1
5
⎢ ⎝⎠ ⎝ ⎠ ⎥ ⎣ 5 5 ⎦ ⎝⎠
⎛ 4 4 4 4 ⎞ 4 1024 2101
= −1 ⎜ ×××× ⎟ = − =
1
⎝ 5 5 5 5 ⎠ 5 3125 3125
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