STATISTICS - Section 1

 

Probability 1

 

 

Basic Concepts

Set Notation

Venn Diagrams

 

 

 

 

Basic Concepts

 

Probability(P) is a number falling within the limits 0≤ P ≤1 .

 

It is described as 'the chance of a particular outcome happening'.

 

 

 

 

The probability of an event 'C' occuring when the outcome is certain (ie there is no other outcome) is 1.

 

This is written:

 

 

However, an event 'I' which is impossible has a probability of 0.

Similarly, this is written:

 

 

 

The probability of an event 'N' not happening is given by:

 

 

'N' not happening = 1 - (probability of it happening 'H')

 

 

In 'probability format' this is:

 

 

 

The sum of probabilities is always 1.

 

 

 

 

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Set Notation

 

If the probability of one event X happening is P(X),

 

and

 

the probability of a different event Y happening is P(Y).

 

 

Then the probability of both events X AND Y happening is given by:

 

 

 

 

The probability of events X OR Y happening is given by:

 

 

 

 

 

The complement of an event is the probability of the event NOT HAPPENING.

 

This is written with an apostrophe after the 2nd bracket.

 

So the probability of event X not happening is written: P(X)' .

 

Adding the probabilities of the event happening and the event not happening gives 1.

 

 

 

 

This means that the event P(X) OR the event P(X)' will happen for certain.

 

Alternatively,

 

 

 

 

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Venn Diagrams

 

Say P(C) is the probability that students in a college choose chemistry.

 

So out of the whole college population Ntotalonly a certain number Nchemistry (red dots) will make this choice.

 

The probability P(C) is given by:

 

 

 

 

Likewise if students choose biology (blue dots), the probability P(B) of this is given by :

 

 

 

 

The total number of elements in a Venn diagram is represented by the sample space.

 

This is a rectangle. So all the students in the college( Ntotal) are contained within this space.

 

 

sample space

 

 

A Venn diagram is a way of ordering choices.

 

The probabilities P(C) and P(B) can be represented by circles within the rectangle, coralling the choices.

 

 

Venn diagram #2

 

 

Now consider the case of students choosing both chemistry AND biology. This would be represented by an overlap of the two circles.

 

 

venn diagram #3

 

 

Then each group representing a choice of ' just chemistry' or 'just biology' would be the respective circles with a lens shaped area missing.

 

We can now write down an expression in set notation from the diagram.

 

 

  full blue circle + full red circle

 

=

 

 overlaps + (incomplete red circle + incomplete blue circle)

 

 

 

Venn diagram #4

 

 

 

 

In words,

 

the probability of students choosing chemistry plus the probability of students choosing biology

 

=

 

the probability that students choose chemistry and biology plus the probability that students choose chemistry or biology

 

 

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Example

 

Out of a group of 45 students, 17 are taking chemistry and 22 biology.

 

i) If 9 students do both chemistry and biology:

 

a) How many students are in neither class?

b) How many are in either class?

 

ii) What is the probability that of a student from the group is taking only the chemistry class? (ans, 3 d.p.)

 

 

Answer:

 

Venn diagram #5

 

 

If only 9 students do chemistry & biology, then the number that do only chemistry,

 

   = (total no. chem. students) - (no. that do chem. & biology)


   = 17 - 9 = 8

 

 

Similarly, if 9 students do chemistry & biology, then the number that do only biology,

 

   = (total no. bio. students) - (no. that do chem. & biology)


   = 22 - 9 = 13

 

 

The no. of students doing single chemistry, single biology and both chemistry and biology

 

  = 8 + 13 + 9 = 30

 

Since the total no. of students is 45, those not taking any of these sciences = 45 - 30 = 15

 

 

To sum up,

 

i)

a) The number of students in neither class is 15.

 

b) There are 8 in chemistry and 13 in biology.

 

 

ii) The probability of studying only chemistry is,

 

 

 

 

 

Final answer, to 3 decimal places, the probability is 0.178

 

 

 

 

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