Probability I

7.1    Sample Space

An experiment is a process or an operation with an outcome.

E.g.  1

Toss a balanced dice once and observe its uppermost surface

E.g.  2
 When a coin is tossed, we can only get 2 results :

Ø  Head                      

Ø  Tail                      

The set of all possible outcomes of an experiment is called a sample space.  It is usually denoted by S.

E.g.  3

En. Adam has a fruit stall.  He sells bananas, apples, watermelons, papayas and durians.  Students of 4KP were asked to select their favourite fruit from the variety of fruits sold at En. Adam’s stall.

S = { bananas, apples, watermelons, papayas, durians }

E.g.  4

A month is randomly selected from a year.  Describe the sample space of this experiment by using set notation.

S = { January, February, March, April, June, July, August, September, October, November, December }

7.2    Event

- Is a subset of the sample space.

- Is an outcome or a set of outcomes that satisfies certain conditions. 
- Denoted by a capital letter.

E.g.  1

A box contains five cards written with 1, 2, 3, 4 and 5 respectively.  A card is picked randomly from a box.

S = { 1, 2, 3, 4, 5 }

If we define just “the card with even numbers”, the outcome of J in set notation will be
J = { 2, 4 }

E.g.  2

A letter is randomly selected from the word “COMPUTER”.  Determine the number of the possible outcomes of the event that the selected letter is.

        i.            A vowel

      ii.            A consonant

Solution :

Let A = event that the selected letter is a vowel = { O, U, E }
Therefore, n (A) = 3

Let B = event that the selected letter is a consonant = { C, M, P, T, R }
Therefore, n (A) = 5

7.3    Probability of an Event

Probability of an event E,

- P (E) = ___number of outcomes of the event___
              number of outcomes of the sample space

- P (E) = n (E)
                        n (S)

- 0 ≤ P (E) ≤ 1

- P (E) = 0 means that it is impossible for the event to happen.

- P (E) = 1 means that the event is certain to happen.

- The closer the probability of a given event is to 1, the more likely it is to happen.

E.g.  1

A bag contains 3 red balls and 4 white ones.  If Rashid puts his hand in the bag and picks a ball, what is the probability that the ball he picked is white?

Solution :

S = { R1, R2, R3, W1, W2, W3, W4 }
n (S) = 7

Let E is the event of drawing a white ball
E = { W1, W2, W3, W4 }
n (E) = 4

Therefore, the probability of drawing a white ball is :          4
                                                                                        7

By Melissa Teh

4 Significant Figures

1. Any digit that is not zero is significant. 1234.56 6 significant figures 

1234.56 6 significant figures

2. Zeros between non-zero digits are significant.

1002.5 5 significant figures

3. Zeros to the left of the first non-zero digit are not significant.

000456 3 significant figures

0.0056 2 significant figures

4. If the number is greater than one (1), then all zeros to the right of the decimal point are significant.

457.12 5 significant figures

400.00 5 significant figures

5. If the number is less than one, then only zeros that are at the end of the number and between non-zero digits are significant.

0.01020 4 significant figures

6. For numbers that do not contain decimal points, the trailing zeros may or may not be significant. In this course assume the digits are significant unless told otherwise.

1000 1, 2, 3, or 4 significant figures. UNCLEAR assume 4 in calculation

0.0010 2 significant figures

1.000 4 significant figures

7. Assume defined and counted quantities have an unlimited number of significant figures.

NOTE: It is much easier to count and keep track of significant figures if the number is written in scientific notation.

By Pravetha Nair

Statistics III

1. Data can be grouped into classes such as 1-5, 6-10, 11-15, ...

          Each class is known as a class interval

2. Range of the class interval is the difference between the smallest and the largest values.

    Range of class interval = Upper limit – Lower limit

3. For a class interval such as 1-5, the number 1 is called the lower limit and the number 5 is called the upper limit. 0.5 is the lower boundary and 5.5 is the upper boundary.

4.   4.  Class size or size of the class interval = Upper boundary – Lower boundary

5.   5.  Modal class is the class with highest frequency.

6.   6.  Midpoint of a class = Lower limit + Upper limit
                                                                2

1.    7.  Mean of a grouped data,     where f is its frequency and x is the midpoint of a class. ∑ means ‘sum of’.

8.  Histogram is a graphical representation of a frequency distribution.

1.       Horizontal axis can be labeled with midpoints or class boundaries or class intervals.

9. Frequency polygon is another graphical representation of a frequency distribution.   

10. The cumulative frequency of a data or a class interval in a frequency table is the sum of its frequency  and the total frequency of the entire class interval before it.

11. An ogive or cumulative frequency curve is a graphical representation of a cumulative frequency table.

1.       12. For ungrouped data, Range = Largest value – Smallest value
For example: 2, 4, 1, 8, 10, 7, 13, 5
Range = 13 – 1 = 12
For grouped data, Range = Midpoint of the last class interval – Midpoint of the first class interval
For example:

Mass (g)	4 - 8	9 - 13	14 - 18	19 - 23	24 – 28
Frequency	3	6	12	5	2

Range   = 24 + 28    _    4 + 8
                      2                   2
              = 26 – 6  
              = 20

by: Mawar Ayu