ABSTRACT

The probability of something happening is the likelihood or chance of it happening.Values of probability lie between 0 and 1, where 0 represents an absolute impossibility and 1 represents an absolute certainty. The probability of an event happening usually lies somewhere between these two extreme values and is expressed as either a proper or decimal fraction. Examples of probability are

that a length of copper wire has zero resistance at 100◦C 0

that a fair, six-sided dice

will stop with a 3 upwards 1 6

or 0.1667

that a fair coin will land with a head upwards

1 2

or 0.5

that a length of copper wire has some resistance at 100◦C 1

If p is the probability of an event happening and q is the probability of the same event not happening, then the total probability is p+ q and is equal to unity, since it is an absolute certainty that the event either will or will not occur; i.e. p+ q = 1

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 probability of selecting at random a man, p, is given by the ratio

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 woman, q , is given by the ratio

number of women number in crowd

i.e. q = 33 20+ 33 =

33 53

or 0.6226

the

p = 20 53

and q = 33 53

, thus the total probability,

p+ q = 20 53

+ 33 53

= 1

hence no obvious error has been made.)

The expectation, E , of an event happening is defined in general terms as the product of the probability p of an event happening and the number of attempts made, n; i.e. E = pn Thus, since the probability of obtaining a 3 upwards when rolling a fair dice is 1/6, the expectation of getting a 3 upwards on four throws of the dice is

1 6

× 4, i.e. 2 3

Thus expectation is the average occurrence of an event.