The Chemistry Maths Book, Second Edition

21.4 Combinations of probabilities 603

The variance Vand standard deviation σof the probability distribution (of the

population) are given by*

(21.13)

(see equations (21.5) to (21.7) for the corresponding quantities for a frequency

distribution).

21.4 Combinations of probabilities

Exclusive events

The two possible outcomes of the toss of a coin are exclusive events; if one happens

then the other does not. Thus, the probability that the outcome is HandTin any one

toss is zero (impossibility), but the probability that the outcome is HorTis unity

(certainty):

P(H or T) 1 = 1 P(H) 1 + 1 P(T) 1 = 11 (21.14)

In general, if Aand Bare exclusive events with probabilitiesP(A)andP(B), the

probability that the outcome is event Aor event Bis

P(A or B) 1 = 1 P(A) 1 + 1 P(B) (exclusive events) (21.15)

EXAMPLE 21.3Exclusive events

The possible results of throws of a die are the numbers {1, 2, 3, 4, 5, 6}, each with

probability (for a fair die).

(i) The probability that the outcome of throwing the die is greater than 4 (is 5 or 6) is

(ii) The probability that the outcome is even is

0 Exercises 7, 8

PP( ) (= 2462463 ) () () ()even=++=×=PPP or or

1

6

1

2

PP PP() ( ) () ()>= 456562 = + =×=

1

6

1

3

or

1

6

Vx x Px x x

i

k

ii

() ( ) ( )=− ==〉−〈〉

=

∑

1

2222

μσ〈

*The symbol σis always used for the standard deviation of the population, but is sometimes also used instead

of sfor a sample; see Section 21.11.