Basic Theory of Probability Chapter-(i)

Probability is a study of mathematics about chances. When things happen randomly method of probability is used  to find how likely a particular event will occur. Just imagine a coin being tossed up in the air and the gravity of earth will bring it back to the ground and now the coin will land on its one of the flat surfaces, but we don't know for sure which side of the flat surfaces of it is going to be in contact with the ground or which side of it is going to be visible to us, either head or tail in both ways. Nevertheless, as a matter of fact there are equal chances for head or tail landing face up provided that the coin is in pristine condition. What we all know for sure is that there are only two ways the coin would have landed either with head facing up or tail facing up. Since these two outcomes are the only possible total number of outcomes, we call these outcomes as members of the SAMPLE SPACE, which consist of all possible outcomes from an occurrence. 
(Someone could think there are also possible other two outcomes at which, it could have landed on its curved surface standing straight up, or it could not have fallen back at all by escaping from the pull of gravity. For the sake of argument both possibilities are valid and countable. However, these two outcomes have a very remote chance to happen, either of which, we call a Zero Probability, which we will discuss little later.) 

Despite above all being said about the probability I just want to mention some thing to rectify the wrong concept that some readers might have about the study of probability. Probability can't predict anything. However, another branch of mathematics known as statistics employs probability along with other several methods of its own to predict future outcomes like for an example, behavior of stock market or unemployment rate in the future based upon the data obtained from various sources.

Probability Experiments and Sample Space:  A sample space is populated  with outcomes obtained from repeated  probability experiments.
Example; tossing a coin several time and writing down the outcomes. An outcome is a result obtained from a test trial. Following table shows a sample space from two different probability experiments coin tossing and dice rolling.

Experiment Sample Space
Toss of a coin Head (H), Tail (T)
Roll of a die 1, 2, 3, 4, 5, 6

Or you can represent them in a Venn Diagram, since  outcomes of an occurrence can be considered a set.

Each outcome from the sample space is called a member or an element.

Classic Probability: Let's come back to the discussion of coin tossing, and an outcome, at which our coin landed showing its face up. We call this is an event of getting a head from coin toss and obviously the other event is getting a tail.
Therefore,
the Sample Space from Coin tossing is


{H,T} H for Head and T For Tail.

Probability to get a head is 1/2, because probability is defined by this following formula

P (E)  =
Number of favorable outcomes
________________________
Total number of outcomes

Where P(E) stands for the "Probability of an event". To calculate the probability of getting a tail from the coin toss, when a coin is tossed as we discussed there are two outcomes, now look at our sample space, which is {H,T} where one chance to get a tail while total outcomes are two, now by applying the probability formula.

P (Tail)  =

    
1
-
2
 

Multiple coin tossing: When two different coins are tossed simultaneously as you can expect there are total of 4 outcomes. Our sample space in this case will be {HH, HT, TH, TT}


Number of possible outcomes can also be mathematically computed without preparing a sample space, which is  2x2 = 4. Likewise when 3 coins are tossed number of total outcomes will be 2x2x2 = 8. If n coins are tossed total members in the sample space will be 2n. That is the total number of outcomes when n coins are tossed.

Rolling a Die: As we know a die (singular form of dice) is a cubic solid object with six faces each of which numbered from 1 to 6. Therefore, when a die is rolled getting a number is decided by the face showing the number upward. As you know it could be any number between 1 and 6 including them. Therefore,, our sample space from a die rolling is

{1, 2, 3, 4, 5, 6}

There are 6 members in our sample space. I.e. Total number of outcomes are 6, but chance to get any number is 1 out of 6, thus the probability of getting, let's say number 3 is 

P (3)  =

    
1
-
6
 

Empirical Probability:
Probabilities can be computed for situations that do not use sample space. In that case frequency distributions are used  and the probability is called Empirical probability.

Example: Imagine a class where,

Freshmen 4
Sophomore8
Juniors6
Seniors7
Total25

Above shown table is an example for a frequency distribution. Probability of an event E can be given by the following formula.

P (E)  =
Frequency of E
____________________
Sum of frequency

Example: To calculate the probability to pick a junior from above distribution, we may use the above formula

Frequency of juniors=6 and sum of frequency=25; therefore the probability to pick a junior from the class is 

P (Junior)  =

    
6
--
25
 

Probability Rules: Probability should be given in a number between 0 and 1 or from 0% to 100%. Let's say probability of an event is P(E) then


If P (E) = 0 then the event E is not likely to happen at all and when P (E) = 1 then the event E will most likely to happen. 

And if probability of an event to happen is P (E) then probability that event will not happen is 1-P (E).

Examples for zero probability is a tossed up coin never falls to the ground or getting a 7 from a die and on the other hand, an example for the probability being =1, is an event of getting either head or tail from a coin toss.

A real world problem (1)

Basket of balls

Shown above was a basket having 3 red balls, 11 green balls, and 7 blue balls that is total of (3 + 11 + 7) = 21 balls altogether. Now we do some exercise on some classical probability problems.
Sorry about the poor illustration
. I know it's not up to the standards, nevertheless, you have to excuse me since I am not an artist.

  1. Find the probability of selecting a red ball without looking in the basket 
  2. Find the probability of selecting a green ball without looking in the basket
  3. Find the probability of selecting a blue ball without looking in the basket

Answers: We assume that you place the ball back in to the basket after each pick. 

  1. Questions is what are the chances of getting a red ball, there are total of 21 balls. Out of 21 balls there are 3 red balls. Now we know total possible outcomes = 21, and Favorable outcome = 3. Use the formula that we learnt about directly to get the answer

    P (E)  =
    Number of favorable outcomes
    ____________________________
    Total number of outcomes

    Therefore, the answer is

    P(Red)  =

        
    3
    --
    21
     
     

    Now you may try to find the answers for questions 2 and 3 exactly the same way I solved the problem 1.

Drawing a card from a deck: As you know a deck of cards has 52 cards. There are four different kinds of suites namely Heart, Diamond, Club, and Spades. While Hearts and Diamonds cards are red in colors, Club and Spades are black. Each kind has 13 different cards. For an example Diamond has 13 different cards whose numbers are staring from A(ce) then 2, 3, 4, 5, 6, 7, 8, 9, 10, J(ack), Q(ueen) and K(ing). Likewise, Hearts, Clubs and Spades also have their own sets of 13 cards. That gives us total of 52 cards since 4 x 13 = 52. Therefore, out of 52 cards, we know there are 26 black cards and 26 red cards in a deck, and there should be four K's, four Q's ... and four 2's and so on. We also know that there are two black K's and two red K's. All the J's, K's and Q's are printed with a human face, so there are total of 12 cards have face on them of which, 6 are black and the other 6 are red. These are the few basic facts we should know about a deck of cards before trying to solve a probability question involving drawing a card from a deck.

A real world problem (2)

Sample Questions:
Imagine you are given a deck of card. We assume that all the cards have equal chance to be picked and they are all look the same when they are placed face down. 

  1. Find the probability of drawing a Diamond King from the deck.
  2. Find the probability of drawing a Club Ace from the deck.
  3. Find the probability of drawing a Heart 9 from the deck.
  4. Find the probability of drawing any red from the deck.
  5. Find the probability of drawing any black King from the deck.
  6. Find the probability of drawing a Club Jack from the deck.
  7. Find the probability of drawing a any Spades from the deck.

We will solve the problem 1 now. We have to find the probability of drawing a Diamond K out of 52 cards. Out of 52 cards there is only one Diamond King, therefore, Number of favorable outcome is 1, but Number of total possible outcomes =52, so the probability is

P(Diamond King)=
1
--
52

Now problem 2 and 3 are very similar to problem 1 so I leave for you to answer.

We try to solve problem 4 now. We need to find the probability of picking a red card from the deck. Since we know that there are 26 red cards in a deck, so Number of favorable outcomes = 26 and Number of Total possible outcomes = 52 therefore the probability is

P (Red Card)=
26
--
52

We have to give the answer in the smallest possible fraction. Therefore the answer is

P (Red Card)=
1
--
2

Probability also can be given in percentage, like a weather man gives a forecast about a rain shower next day. He would usually say tomorrow there is a 30% chance of rain. Likewise we can also say in case of preceding example that the probability of a red card being drawn is 50%.

Law of large numbers: As we have learned already probability can't predict anything, I don't want any of the readers of this article to think that when coin is tossed every other time the result would be same. What probability suggest is that the chance of getting either head or tail is same. However, lets say a coin is tossed 1000 times, having a head or tail is close to being 500. If you increase the number of test by many fold having a head or tail for equal number of time becoming closer and closer.

Next Chapter

Visit My Website