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High School Mathematics - 2 Sample Space

Sample space

Set of all possible outcomes of an experiment or trial is called sample space and is denoted by S

Consider the following examples

1. Tossing of a coin
``` The outcome will be either heads or tails
so S = {H , T}
n(S) = 2 ```
2. Tossing of 2 coins
```If 2 coins are tossed each can give two results
so  S = { HT , HH , TT, TH}
n(S) = 4 ```
3. Tossing of a dice
```     S = { 1,2,3,4,5,6 }
n(S) = 6 ```

Events

Consider example 2 stated above
Now we may be interested in some happening. Consider the outcome that the pair has one heads and one tails
so E = { HT , TH ) so n(E) = {2}
Thus the chance that an event happens is called Probability
Formula P(E) = n(E) / n(S)

Solved Examples

1. 7 books on physics and 4 on mathematics are to be arranged at random on a shelf. Find the probability that the books on maths do not stand next to each other

Solution

11 books can be selected in 11P11 ways
n(S) = 11!
Let B be the event that no 2 maths book stand next to each other
Look at the following arrangement
M P M P M P M P M P M P M P M
7 books on physics can be arranged in 7P7= 7!ways
The 4 maths books can take any of the 8 positions shown above.
n(B) = 7P7x8P4
P(B) = n(B)/n(S) = 7!x8x7x6x5/11!
P(B) = 7/33

2. The letters of the word 'E Q U A T I O N' are arranged at random . Find the probability that the arrangement
a)Starts with a vowel and ends with a consonant.
b)Starts with a vowel and ends with a vowel 5/14.

Solution

n(s) = 8P8= 8!
Consider the first case(a)
Let A be the event that the word starts with a vowel and ends with a consonant.
As there are 5 vowels in the word the first position can be filled in 5P1=5 ways.
As there are 3 consonants in the word the last position can be filled in 3P1=3 ways.
So far we have taken care of 2 positions.
* - - - - - - *
Now the 6 spaces inbetween can be filled in 6P6=6! ways.
So n(A) = 5x6!x3
So P(A) = 5x6!x3/8! = 15/56.
Consider the second case (b)
Let B be the event that the word starts with a vowel and ends with a vowel.
As there are 5 vowels in the word the first position can be filled in 5P1=5 ways.
With 4 vowels remaining the last the last position can be filled in sup>4P1=4 ways.
* - - - - - - *
Now the 6 spaces inbetween can be filled in 6P6=6! ways.
So n(B) = 5x6!x4/8!
So P(B) = 5x6!x4/8!

3. Tom and Dick play with 2 dice. If Tom throws 9 what is Dick's chance of throwing a higher number?

Solution

Total number of outcome of throwing 2 dice = 6x6 = 36. Dick must throw either 10,11,12. This can be done in 5 ways.
(4+6,5+6,6+4,6+5,6+6)
Dicks chance of throwing a higher number is 5/36.

4. A 3 digit number is to be formed by using the digits 1,2,3,4,5,6,7,8,9.(Repetions not allowed). What is the probability that the digit formed is greater than 600.

Solution

The possibility of forming 3 digit numbers is 9P3 = 504. n(S)= 504. Let A be the event that the number is greater 600.
n(A)= 4x8P2 = 224.
P(A)= 224/504.

5. A bag contains 3 white , 4 black and 2 red balls. 2 balls are chosen at random. What is the probability that 1 white and 1 red ball is chosen.

Solution

2 balls can be drawn from 9 balls in 9C2 ways = 36 ways n(S)= 36 n(A)= 3C1.2C1 = 6 P(A) = 6/36 = 1/6