Probabilty

If an event can happen in 'a' ways and fall in 'b' ways, and each of these ways is equally likely, then probability or the chance, of its happening is , and that of its failing is . E.g., If in a lottery there are 7 prizes and 25 blanks, the chance that a person holding 1 ticket will win a prize is , and his chance of winning is .
If p is the probability of happening of an event, the probability of its not happening is 1 - p.

Instead of saying that the chance of the happening of an event is , it is sometimes stated that the odds are 'a' to 'b' in favour of the event, or 'b' to 'a' against the event.
If c is the total no. of cases, each being equally likely to occur, and of these 'a' are favourable to the event, then the probability that the event will happen is , and the probability that it will not happen is 1 - .

Example: What is the chance of throwing a number greater than 4 with an ordinary dice whose faces are numbered from 1 to 6.

There are 6 possible ways in which the dice can fall, and of these two are favourable to the event required.
required chances =

Example: From a bag containing 4 white and 5 black balls a man draws 3 at random, what are the odds against these being all black?

The total no. of ways in which 3 balls can be drawn is 9C3 and the no. of ways of drawing 3 black balls is 5C3;

therefore the chance of drawing 3 black balls is

Thus the odds against the event are 37 to 5.

Example: Find the chance of throwing at least one ace in a simple throw with two dice.

The possible no. of cases is 6 × 6 or 36.
An ace on one die may be associated with any of the 6 numbers on the other die, and the remaining 5 numbers on the first die may be associated with the ace on the second die, thus the number of favourable cases is 11.
required chance = .