Simple Probability

 
  • Probability is the measurement of the possible occurrence of an event expressed either in the form of fractions or percentages.
 
13.1  Experimental Probability
 
  Definition  
     
 

The probability that is obtained from an experiment.

 
 

In general,

\(\begin{aligned} &\space\text{The experimental probability of an event} \\\\&=\dfrac{\text{Frequency of an event}}{\text{Number of trials}} \end{aligned}\)

 
 
13.2  The Probability Theory Involving Equally Likely Outcomes
 
Sample space for an experiment:
 
  Definition  
     
 
  • Sample space is a set of all possible outcomes of an experiment.
  • It is represented with the letter \(S\).
 
 
  • A tree diagram can be used to show the flow of the process.
  • It is used to organise and calculate the probability of an event happening.
   
Example
   
   
Event of an experiment:
 
  Definition  
     
 

Event is a set of possible outcomes that fulfill certain conditions for a sample space and is a subset for the sample space.

 
 
Probability of an event:
 
  • The number of possible outcomes is represented by \(n(S)\) .
  • The number of events is represented by \(n(A)\).
  • The probability of an event \(A\) is \(P(A)\).
 

Then, the probability of an event \(A\) is represented by 

\(P(A) = \dfrac{n(A)}{n(S)}.\)

 
13.3  Complement of An Event Probability
 

The probability for the complement of an event, \(P(A')\)

\(\begin{aligned} P(A) + P(A') &= 1 \\\\ P(A') &= 1- P (A) \\\\ \end{aligned}\)

where \(0 \le P(A) \le 1\).

 
13.4  Simple Probability
 
  Example  
     
 

A box contains \(8\) green balls, \(12\) black balls and a number of white balls.

A white ball is chosen at random from the box.

The probability of getting a white ball is \(\dfrac{3}{7}\)‚Äč.

What is the number of white balls in the box?

 
     
 

Let \(w\) be the number of white balls.

Then,

\(\begin{aligned}n(S)&=8+12+w\\\\&=20+w.\\\\\end{aligned}\)

So,

\(\begin{aligned} P(w)&=\dfrac{n(w)}{n(S)} \\\\\dfrac{3}{7}&=\dfrac{w}{20+w} \\\\3(20+w)&=7(w) \\\\60+3w&=7w \\\\4w&=60 \\\\w&=15.\\\\ \end{aligned}\)

Thus, the number of white balls in the box is \(15\).