Grade 10 Maths Chapter 15 - Probability

What is Probability ?

• Probability is the study of mathematics which calculates the degree of uncertainty. 
• The chance that an event will or will not occur is expressed on a scale ranging from 0-1.
• It can also be represented as a percentage, where 0% denotes an impossible event and 100 % implies a certain event.
  

 Event and Outcome

• An Outcome is a result of a random experiment. For example, when we roll a dice getting six is an outcome.
   
• An Event is a set of outcomes. For example when we roll dice the probability of getting a number less than five is an event

Note: An Event can have a single outcome.

 There are two types of approaches to study probability-

1.  Experimental or Empirical Probability
2.  Probability — A Theoretical Approach

Experimental or Empirical Probability

•  Experimental probability can be applied to any event associated with an experiment that is repeated a large number of times.
• A trial is when the experiment is performed once.
• The result of probability based on the actual experiment is called experimental probability.
• It is also known as empirical probability.
  
• In this case, the results could be different if we do the same experiment again.
• Experimental or empirical probability: P(E) =Number of trials  where the event occurred/Total Number of Trials
  

Probability — A Theoretical Approach

• Here we assume that the outcomes of the experiment are equally likely.
  
• In the theoretical approach, we predict the results without performing the experiment actually. 
• The other name of theoretical probability is classical probability.
• Theoretical Probability, P(E) = Number of Outcomes Favourable to E / Number of all possible outcomes of the experiment
  

Where the outcomes are equally likely.

Equally Likely Outcomes

• If we have the same possibility of getting each outcome then it is called equally likely outcomes. 
• Example - A dice have the same possibility of getting 1, 2, 3, 4, 5 and 6.
  

Not Equally Likely

• If we don't have the same possibility of getting each outcome then it is said to be the not equally likely outcome.
• Example - 3 green balls and 2 pink balls are not equally likely as the possibility of the green ball is 3 and the possibility of the pink ball is 2.
  

Elementary Event

• If an event has only one possible outcome of the experiment then it is called an elementary event.
• Example:  

Take the experiment of tossing a coin n number of times. One trial of this experiment has two possible outcomes: Heads(H) or Tails(T).

 

So for an individual toss, it has only one outcome, i.e Heads or Tails. 

Sum of Probabilities

• The sum of the probabilities of all the elementary events of an experiment is one.
• Example: take the coin-tossing experiment. 

P(Heads) + P(Tails )

= (1/2)+ (1/2) =1

The sum of the probabilities of all the elementary events of an experiment is 1.

Impossible Events

• If there is no possibility of an event to occur then its probability is zero. This is known as an impossible event. 
•   i.e. P(E) = 0.
• Example -

1.  It is not possible to draw a green ball from a group of blue balls.
2.  Probability of getting a 7 on a roll of a die is 0. As 7 can never be an outcome of this trial

Sure or Certain Event

• An event that has a 100% probability of occurrence is called a sure event. 
• The probability of occurrence of a sure event is one.
  
• If the possibility of an event to occur is sure then it is said to be the sure probability. Here the probability is one. 
• Example -  What is the probability that a number obtained after throwing a die is less than 7?
  So,  P(E) = P(Getting a number less than 7) = 6/6= 1 

Range of Probability of an event

The range of probability of an event lies between 0 and 1 inclusive of 0 and 1, i.e.

 0  ≤  P(E)  ≤  1.

Geometric Probability

• Geometric probability is the calculation of the likelihood that one will hit a particular area of a figure.
•  It is calculated by dividing the desired area by the total area.
•  In the case of Geometrical probability, there are infinite outcomes.

Complementary Events

• Complementary events are two outcomes of an event that are the only two possible outcomes.
•  This is like flipping a coin and getting heads or tails. 
• P(E)+P(E¯)=1, where E and E¯ are complementary events. The event E¯, representing ‘not E‘, is called the complement of the event E.

 Tossing a Coin

• When a coin is tossed, there are two possible outcomes:

1.   heads (H) or
2.   tails (T)

• We say that the probability of the coin landing H is ½

And the probability of the coin landing T is ½

 Pack of Cards

A pack of cards has a total of 52 cards, and those are segregated in four suits –

•     Spades (Black coloured)
•     Hearts (Red coloured)
•     Diamond (Red coloured)
•     Clubs (Black coloured)

Now, each suite contains 13 cards starting right from 1 to 13. However, fewer cards are named differently.

    First Card - Termed as an ‘Ace’ and has a symbol of ‘A’ followed by a series of numbers starting from 2 to 10.

    11th card - Termed as ‘Jack’ and indicated by a symbol ‘J.’

    12th card - The ‘Queen’ and indicated with a symbol ‘Q.’

    13th card - A ‘King’ which has a symbol ‘K’ on it.

    Jack, Queen, and King - Called face cards. Therefore, there are 12 face cards in a pack of 52 cards. Utilising this number, you can calculate the probability of an event’s occurrence.

    Since the spades and clubs are black coloured and hearts and diamonds are red coloured, there are 26 black cards and 26 red cards. 

Probability of Dice

• When we throw a dice , there are six possible outcomes:

            { 1 , 2 , 3 , 4 , 5 , 6 }

• Die has six faces with each face representing a number, starting from 1 to 6. So, the outcome of occurrence of any of the number starting from 1 up until 6 is equal.  

• It means that the probability of occurrence of 1 on a throw of dice is the same as the probability of occurrence of 4 on the throw of a dice. It stands true for any other number between 1 to 6. The probability for the occurrence of each such event will be 1/6.

• Further, you will have to understand and analyse the number of such outcomes in an experiment of throwing two dice or three. You will have to analyse the number of such possible events in a random trial. 

Some Solved Examples

Example: 1

What is the probability of drawing a heart from a deck of cards?

Solution:

We know that there are total 52 cards in a deck out of which 13 cards are of heart.

So the favourable outcomes are 13 and the total no. of events is 52.

Favorable Outcome = 13/52 = 1/4

 

Example: 2

If we toss two coins together, then what is the probability of getting at least one tail?

Solution:

If we toss two coins together then the total outcomes could be

Total outcomes

The favorable outcomes for at least one head will be

{HH}, {HT}, {TH} = 3

P (for at least one head) = 3/4