Higher Engineering Mathematics, Sixth Edition

Probability 549

56.2 Laws of probability

The addition law of probability

The addition law of probability is recognized by the
word‘or’joining the probabilities. IfpAis the proba-
bility of eventAhappening andpBis the probability
of eventBhappening, the probability ofeventAor
eventBhappeningis given bypA+pB(providedevents
AandBaremutuallyexclusive,i.e.AandBare events
which cannot occur together). Similarly, the probability
of eventsAorBorCor...Nhappening is given by

pA+pB+pC+···+pN

The multiplicationlaw of probability

The multiplication law of probability is recognized by
the word‘and’joining the probabilities. IfpAis the
probability of eventAhappening andpBis the proba-
bility of eventBhappening, the probability ofeventA
and eventBhappening is given bypA×pB. Similarly,
theprobabilityofeventsAandBandCand...Nhap-
pening is given by

pA×pB×pC×···×pN

56.3 Workedproblemsonprobability

Problem 1. Determine the probabilities of

selecting at random (a) a man, and (b) a woman

from a crowd containing 20 men and 33 women.

(a) The probabilityof selecting at random a man,p,is

givenbytheratio

number of men

number in crowd

i.e. p=

20

20 + 33

=

20

53

or 0. 3774

(b) The probability of selecting at random a women,

q, is given by the ratio

number of women

number in crowd

i.e. q=

33

20 + 33

=

33

53

or 0. 6226

(Check: the total probabilityshould be equal to 1;

p=

20

53

andq=

33

53

thus the total probability,

p+q=

20

53

+

33

53

= 1

hence no obvious error has been made).

Problem 2. Find the expectation of obtaining a 4

upwards with 3 throws of a fair dice.

Expectation is the average occurrence of an event and is

defined as the probabilitytimes the number of attempts.

The probability,p, of obtaining a 4 upwards for one

throw of the dice is^16

Also, 3 attempts are made, hencen=3andthe

expectation,E,ispn,i.e.E=^16 × 3 =^12 or 0. 50

Problem 3. Calculate the probabilities of

selecting at random:

(a) the winning horse in a race in which 10 horses

are running,

(b) the winning horses in both the first and second

races if there are 10 horses in each race.

(a) Since only one of the ten horses can win, the prob-

ability of selecting at random the winning horse is

number of winners

number of horses

,i.e.

1

10

or0.10

(b) The probability of selecting the winning horse in

the first race is 101. The probability of selecting

the winning horse in the second race is 101 .The

probability of selecting the winning horses in the

firstandsecond race is given by the multiplication

law of probability, i.e.

probability=

1

10

×

1

10

=

1

100

or 0. 01

Problem 4. The probability of a component

failing in one year due to excessive temperature is

1

20

, due to excessive vibration is

1

25

and due to

excessive humidity is

1

50

. Determine the
probabilities that during a one-year period a
component: (a) fails due to excessive temperature
and excessive vibration, (b) fails due to excessive
vibration or excessive humidity, and (c) will not fail
because of both excessive temperature and
excessive humidity.