PROBABILITY

Afrin Sultana Chowdhury 1

Assistant professor
Dept. of Biotechnology & Genetic Engineering
Noakhali Science & Technology University
afrin.bge@nstu.edu.bd
PROBABILITY
Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random
event. The value is expressed from zero to one. Probability has been introduced in Maths to predict how
likely events are to happen. The meaning of probability is basically the extent to which something is likely
to happen. This is the basic probability theory, which is also used in the probability distribution, where
you will learn the possibility of outcomes for a random experiment. To find the probability of a single
event to occur, first, we should know the total number of possible outcomes.

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total
certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it.
Probability can range in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a
certain event.

For example, when we toss a coin, either we get Head OR Tail, only two possible outcomes are possible
(H, T). But if we toss two coins in the air, there could be three possibilities of events to occur, such as both
the coins show heads or both show tails or one shows heads and one tail, i.e.(H, H), (H, T),(T, T).

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PROBABILITY FORMULA
The probability formula is defined as the possibility of an event to happen is equal to the ratio of the
number of favorable outcomes and the total number of outcomes.

Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes

Example:

There are 6 pillows in a bed, 3 are red, 2 are yellow and 1 is blue. What is the probability of
picking a yellow pillow?

Ans: The probability is equal to the number of yellow pillows in the bed divided by the total number of
pillows, i.e. 2/6 = 1/3.

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Example: There is a container full of colored bottles, red, blue, green and orange. Some of the
bottles are picked out and displaced. Amit did this 1000 times and got the following results:

No. of blue bottles picked out: 300

No. of red bottles: 200
No. of green bottles: 450
No. of orange bottles: 50

a) What is the probability that Amit will pick a green bottle?

Ans: For every 1000 bottles picked out, 450 are green.
Therefore, P(green) = 450/1000 = 0.45

b) If there are 100 bottles in the container, how many of them are likely to be green?
Ans: The experiment implies that 450 out of 1000 bottles are green.
Therefore, out of 100 bottles, 45 are green.

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Probability Tree:

The tree diagram helps to organize and visualize the different possible outcomes. Branches and ends
of the tree are two main positions. Probability of each branch is written on the branch, whereas the
ends are containing the final outcome. Tree diagrams are used to figure out when to multiply and when
to add. You can see below a tree diagram for the coin:

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Applications of Probability:
Probability has a wide variety of applications in real life. Some of the common applications which we see
in our everyday life while checking the results of the following events:

• Choosing a card from the deck of cards

• Flipping a coin
• Throwing a dice in the air
• Pulling a red ball out of a bucket of red and white balls
• Winning a lucky draw

Other Major Applications of Probability

• It is used for risk assessment and modelling in various industries

• Weather forecasting or prediction of weather changes
• Probability of a team winning in a sport based on players and strength of team
• In the share market, chances of getting the hike of share prices

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PROBABILITY RULES
1.) The Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)
If A and B are mutually exclusive events, or those that cannot occur together, then the third term is 0,
and the rule reduces to P(A or B) = P(A) + P(B). For example, you can't flip a coin and have it come up
both heads and tails on one toss.

2.) The Multiplication Rule: P(A and B) = P(A) * P(B|A) or P(B) * P(A|B)
If A and B are independent events, we can reduce the formula to P(A and B) = P(A) * P(B). The term
independent refers to any event whose outcome is not affected by the outcome of another event. For
instance, consider the second of two coin flips, which still has a .50 (50%) probability of landing heads,
regardless of what came up on the first flip. What is the probability that, during the two coin flips, you
come up with tails on the first flip and heads on the second flip?
Let's perform the calculations: P = P(tails) * P(heads) = (0.5) * (0.5) = 0.25

3.) The Complement Rule: P(not A) = 1 - P(A)

Do you see why the complement rule can also be thought of as the subtraction rule? This rule builds
upon the mutually exclusive nature of P(A) and P(not A). These two events can never occur together,
but one of them always has to occur. Therefore P(A) + P(not A) = 1. For example, if the weatherman
says there is a 0.3 chance of rain tomorrow, what are the chances of no rain?
Let's do the math: P(no rain) = 1 - P(rain) = 1 - 0.3 = 0.7
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Mutually Exclusive Events:
In statistics and probability theory, two events are mutually exclusive if they cannot occur at the same
time. The simplest example of mutually exclusive events is a coin toss. A tossed coin outcome can be
either head or tails, but both outcomes cannot occur simultaneously.

Complementary Events:
Two events are said to be complementary when one event occurs if and only if the other does not. The
probabilities of two complimentary events add up to 1.

Independent events do not affect one another's probability of occurring. For example, if I roll a
standard six-sided die and flip a coin, the two events will not have any effect on the probability of the
other. Regardless of the outcome of rolling the die, the coin will be just as likely to land on heads or
tails. Likewise, regardless of the outcome of the coin flip, the die will be just as likely to land on one of
the six numbers of the die. 8