INDEPENDENT AND DEPENDENT EVENTS - GRADE 11

Use of a Venn Diagram

We can also use Venn diagrams to check whether events are dependent or independent.

	Definition: Independent events Events are said to be independent if the result or outcome of the event does not affect the result or outcome of another event. So P(A/C)=P(A), where P(A/C) re presents the probability of event A after event C has occured.
	Definition: Dependent events If the outcome of one event is affected by the outcome of another event such that P(A/C) ≠ P(A)

Also note that For example, we can draw a Venn diagram and a contingency
table to illustrate and analyse the fol lowing example .

Worked Example 154: Venn diagrams and events Question: A school decided that it’s 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? Answer Step 1 : Draw a Venn diagram Step 2 : Draw up a contingency table   Beige Not Beige Totals Blue Not Blue 0.15 0.25 0.4 0.2 0.55 0.35 Totals 0.40 0.6 1 Step 3 : Work out the probabilities P(Blue)=0.4, P(Beige)=0.55, P(Both)=0.15, P(Neither)=0.20 Probability of choosing beige after blue is: Step 4 : Solve the problem Since P(Beige/Blue) ≠ P(Beige) the events are statistically independent.

Extension: Applications of Probability Theory

Two major applications of probability theory in everyday life are in risk assessment
and in trade on commodity markets. Governments typically apply probability meth-
ods 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 con sumer 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.

34.3 End of Chapter Exercises

1. In each of the following contingency tables give the expected numbers for the events to
be perfectly independent and decide if the events are independent or dependent.

A

B

C

 D

2. A company has a probability of 0.4 of meeting their quota on time and a probability of
0.25 of meeting their quota late. Also there is a 0.10 chance of not meeting their quota
on time. Use a Venn diagram and a contingency table to show the information and decide
if the events are independent.

3. A study was undertaken to see how many people in Port Elizabeth owned either a Volk-
swagen or a Toyota. 3% owned both, 25% owned a Toyota and 60% owned a Volkswagen.
Draw a contingency table to show all events and decide if car ownership is independent.

4. Jane invested in the stock market. The probability that she will not lose all her money is
1.32. What is the probability that she will lose all her money? Explain.

5. If D and F are mutually exclusive events, with P(D’)=0.3 and P(D or F)=0.94, find P(F).

6. A car sales person has pink, lime-green and purple models of car A and purple, orange and
multicolour models of car B. One dark night a thief steals a car.

A What is the experiment and sample space?
B Draw a Venn diagram to show this.
C What is the probability of stealing either model A or model B?
D What is the probability of stealing both model A and model B?

7. Event X’s probability is 0.43, Event Y’s probability is 0.24. The probability of both occuring
together is 0.10. What is the probability that X or Y will occur (this inculdes X and Y
occuring simultaneously )?

8. P(H)=0.62, P(J)=0.39 and P(H and J)=0.31. Calculate:

A P(H’)
B P(H or J)
C P(H’ or J’)
D P(H’ or J)
E P(H’ and J’)

9. The last ten letters of the alphabet were placed in a hat and people were asked to pick
one of them. Event D is picking a vowel, Event E is picking a consonant and Evetn F is
picking the last four letters. Calculate the following probabilities:

A P(F’)
B P(F or D)
C P(neither E nor F)
D P(D and E)
E P(E and F)
F P(E and D’)

10. At Dawnview High there are 400 Grade 12’s. 270 do Computer Science, 300 do English and
50 do Typing. All those doing Computer Science do English, 20 take Computer Science and
Typing and 35 take English and Typing. Using a Venn diagram calculate the probability
that a pupil drawn at random will take:

A English, but not Typing or Computer Science
B English but not Typing
C English and Typing but not Computer Science
D English or Typing