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We can also use Venn diagrams to check whether events are dependent or independent.
Events are said to be independent if the result or outcome of one event does not affect the result or outcome of the other event. So P(A/C)=P(A), where P(A/C) represents the probability of event A after event C has occured.
Two events are dependent if the outcome of one event is affected by the outcome of the other event i.e. $P(A/C)\ne P\left(A\right)$
. Also note that $P(A/C)=\frac{P(A\cap C)}{P\left(C\right)}$ . For example, we can draw a Venn diagram and a contingency table to illustrate and analyse the following example.
A school decided that its uniform needed upgrading. The colours on offer were beige or blue or beige and blue. 40% of the school wanted beige, 55% wanted blue and 15% said a combination would be fine. Are the two events independent?
Beige | Not Beige | Totals | |
Blue | 0,15 | 0,4 | 0,55 |
Not Blue | 0,25 | 0,2 | 0,35 |
Totals | 0,40 | 0,6 | 1 |
P(Blue)=0,4, P(Beige)=0,55, P(Both)=0,15, P(Neither)=0,20
Probability of choosing beige after blue is:
Since $P(Beige/Blue)\ne P\left(Beige\right)$ the events are statistically dependent.
Two major applications of probability theory in everyday life are in risk assessment and in trade on commodity markets. Governments typically apply probability methods in environmental regulation where it is called “pathway analysis”, and are often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on statistical analyses of their probable effect on the population as a whole. It is not correct to say that statistics are involved in the modelling itself, as typically the assessments of risk are one-time and thus require more fundamental probability models, e.g. “the probability of another 9/11”. A law of small numbers tends to apply to all such choices and perception of the effect of such choices, which makes probability measures a political matter.
A good example is the effect of the perceived probability of any widespread Middle East conflict on oil prices - which have ripple effects in the economy as a whole. An assessment by a commodity trade that a war is more likely vs. less likely sends prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are not assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict.
It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has had a profound effect on modern society. A good example is the application of game theory, itself based strictly on probability, to the Cold War and the mutual assured destruction doctrine. Accordingly, it may be of some importance to most citizens to understand how odds and probability assessments are made, and how they contribute to reputations and to decisions, especially in a democracy.
Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, utilize reliability theory in the design of the product in order to reduce the probability of failure. The probability of failure is also closely associated with the product's warranty.
Brown eyes | Not Brown eyes | Totals | |
Black hair | 50 | 30 | 80 |
Red hair | 70 | 80 | 150 |
Totals | 120 | 110 | 230 |
Point A | Point B | Totals | |
Busses left late | 15 | 40 | 55 |
Busses left on time | 25 | 20 | 45 |
Totals | 40 | 60 | 100 |
Durban | Bloemfontein | Totals | |
Liked living there | 130 | 30 | 160 |
Did not like living there | 140 | 200 | 340 |
Totals | 270 | 230 | 500 |
Multivitamin A | Multivitamin B | Totals | |
Improvement in health | 400 | 300 | 700 |
No improvement in health | 140 | 120 | 260 |
Totals | 540 | 420 | 960 |
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