Probability without Balls

Probability is the measurement of how often events take place.  It is often taught using coins, dice and bags of balls.  However, it is only football referees and magicians who spend much time messing with these things, but most of us are familiar with catching a bus.  The table below shows the frequency of a bus arriving relative to the scheduled time, it was derived from an imaginary survey conducted from Monday to Friday over a ten week period of a hypothetical bus.

Description	Face	Time	Frequency	Probability
Early		<0.5	5	0.10
On-time		-0.5 to +0.5	14	0.28
Late		>0.5	31	0.62
Total			50	1.00

This table illustrates some aspects of probability.

Estimation of Probability

The probability of an event actually taking place relative to the number of times it could have happened.

For the hypothetical bus, it could have been on time every day, but only managed it on 14 days, thus the probability of it been on time was:

P_{on time} = F_{on time} / N_observations = 14/50 = 0.28

All Outcomes

The sum of the probability of all possible outcomes is one.

In our example, the bus is either early, on time or late.  If we wanted to consider that probability of having to stand or having a seat or the drivers state of mind, we would have to collect some more data.

Mutually Exclusive Events

A single event can only have one outcome, once an outcome has occurred, no other outcome is possible, thus outcomes are mutually exclusive. 

For the hypothetical bus, if it arrived on time, it can't be early or late, being on time, excludes the early or late outcomes.

This leads to two important expressions of probability:

Probability of Mutually Exclusive Outcomes

If the probability of of an outcome is P, then the probability of a mutually exclusive outcome is 1-P.

For our bus, if the probability of the bus being on time is 0.28, then probability of it being early or late is 1.00-0.28=0.72.

Probability of Alternative Outcomes is Additive

If the possible outcomes of an event are A, B, C etc. the probability of the outcome being:

A or B = P(A or B) = P(A)+P(B)

For the bus, the probability of it being early or on time is:

P(early,on-time) = 0.10 + 0.28 = 0.38

Thus the probability of it being late, the mutually exclusive outcome if it is early or on-time is 0.62

Conditional Probability

If the outcome of an event depends on the outcome of a preceding event, the probability of the outcome of a series of events is the product of the probabilities of the outcomes of the successive events. Thus:

P(A and B) = P(A) * P(B)

This describes that the probability of the second event being B, when the outcome of the first event is A.

Back to the bus, its time of arrival each day is an independent event, the probability of a series of a outcomes, such as it being early or on-time on two consecutive days is:

P((early or on-time) and (early or on-time))=0.38*0.38 = 0.14

Sampling without Replacement

The underlying assumption behind the bus example is that the outcome on any given day is independent of that on any other day and that the number of days available for sampling is large.  In the real world, this model is not always realistic.  If the population available for sampling is small, successive samples will reduce the population if the selected item is not replaced.