Probability – Introduction


  • What you will learn in this lesson on Probability?

A bag contains apples and no other fruits. Take out one.

  • What is the chance it is an apple?

The probability is 100% i.e., 1. A bag contains 3 apples and 3 oranges.

  • What is the chance it is an apple?

50%, i.e. 1/2 as each type of fruit has equal chance or probability of being taken out. A dice having six faces with numbers marked from 1 to 6 is tossed.

  • What is the chance that face with number 5 comes up?

1/6, since each numbered-face stands equal chance of coming up

  • In the dice above, what is the chance an even number comes up?

Hmm, needs thought. 2, 4 and 6 are the even numbers on a dice. So, the probability an even number will turn up is 3/6 i.e. ½.

Now, face this scud: Take 2 balls out of a bag in which there are 3 red and 4 green balls.

  • What is the probability that the two balls are red-colored? 

To face this scud, you need to arm yourself with the details of Permutations and Combinations concepts. We will discuss them in light of solving Probability concepts.


PROBABILITY:

Or, if you wish to capture a terse overview of each Probability Formula, then go through each of the following header-links. You can also click the header-links to take you to the page on the specific Probability formula:

DEFINITION OF PROBABILITY:

If E denotes any Event that is required to happen, then probability the event E happens, denoted as P(E) is: (Number of outcomes favorable for the Event E)/ (Total number of Outcomes)

Example:

When a dice is tossed, the probability a prime number appears is 3/6 = ½, P (E) +P (Ê) = 1

Where P (E) denotes the probability that an Event E happens and P (Ê) denotes the probability that the Event E does not happen.
Then: P (E) +P (Ê) = 1

Example:

When a single fair coin is tossed we know: P (H) + P (T) = 1
Let P (E) indicate P (H) and therefore P (T) will indicate P (Ê), then P (H) + P (T) = 1
can be also written as P (E) +P (Ê) = 1; P (AT LEAST ONE) = 1– P(NONE)

Example:

When two unbiased coins are tossed at the same time, the probability that at least one heads will appear is =
1 – P (No Heads will appear) = 1 – ¼ = ¾

INDEPENDENT EVENTS:

Two events A, B are said to be Independent Events if they occur without affecting the probability of occurrence of each other. If two events A, B are independent events then the probability the two events happen at the same time is equal to the multiplication of the respective probability values of the two events, i.e.
P (A and B) = P (A) × P (B)

Example:

When two coins are tossed, the probability that Heads will appear on the first coin and tails on the second is:
P (H and T) = P (H) × P (T) = (½) × (½) = ¼

MUTUALLY EXCLUSIVE EVENTS: Two events A, B are said to be mutually exclusive or just exclusive, if each one of the two events occurs preventing (excluding) the other from taking place at the same time.

Example:

On a single fair coin, each one of the two Heads and Tails happens preventing (excluding) the other from taking place at the same time.

RANDOM EXPERIMENT:

Any activity on doing which the result cannot be certainly stated is called a Random Experiment.

Examples:

  1. Toss an unbiased coin and you can only guess what will happen, not certainly state which of the two Heads, Tails will happen.
    Tossing a coin is therefore a random experiment.
  2. Toss an unbiased dice, you can only guess which of the numbers from 1 to 6 will show up, but not surely bet on one number.
    Tossing a dice is therefore a random experiment

EXHAUSTIVE EVENTS:

What all numbers can you say may occur on tossing a dice? Numbers from 1 to 6. The outcomes, i.e. numbers from 1 to 6 form the list of all outcomes that are likely to happen, so they are collectively called Exhaustive Events (what all are available to occur). On a coin, Heads and Tails form exhaustive events.

EQUALLY LIKELY EVENTS:

Toss an unbiased coin. You have no reason to prefer any one of the two outcomes (Heads, Tails) over the other. So, each one has the same chance of taking place. Two or more events which have the same chance or probability of occurrence are called equally likely events.

Examples:

On tossing an unbiased coin, each of the two outcomes: Heads, Tails have the same chance i.e. probability of occurrence, so they are called equally likely events.
On tossing a fair dice, each of the outcomes, i.e. numbers from 1 to 6 have the same chance i.e. probability of occurrence, so they are called equally likely events.

SAMPLE SPACE:

All the outcomes that are likely to happen when an experiment is done form the set called Sample Space.

Example:

The sample space on tossing a coin is {H, T}
The sample space on tossing a dice is {1, 2, 3, 4, 5, 6}