Presented by:

SANCHIT KUMAR SRIVASTAVM.PHARM

(PHARMACEUTICAL CHEMISTRY)

Topics

IntroductionEmpirical ProbabilityTheoretical ProbabilityCompound EventsAddition Rule

Multiplication Rulefor Dependent EventsMultiplication Rule for Independent EventsBayes TheoremBinomial theoremNormal Distribution

Probability Introduction

When we speak of the probability of something happening, we are referring to the likelihood—or chances—of it happening. Do we have a better chance of it occurring or do we have a better chance of it not occurring?

Generally, we talk about this probability as a fraction, a decimal, or even a percent—• the probability that if two dice are

tossed the spots will total to seven is 1/6

• the probability that a baseball player will get a hit is .273

• the probability that it will rain is 20%

Empirical Probability

Some probabilities are determined from repeated experimentation and observation, recording results, and then using these results to predict expected probability. This kind of probability is referred to as empirical probability.

• If we conduct an experiment and record the number of times a favorable event occurs, then the probability of the event occurring is given by:

We can see this in the following example. If we flip a coin 5 times and it lands on heads 2 times, then the empirical probability is given by:P(HEADS) = 2/5 or 0.4

NOTE

Theoretical ProbabilityOther probabilities are determined using mathematical computations based on possible results, or outcomes. This kind of probability is referred to as theoretical probability.The theoretical probability of event E happening is given by:

Q). If we consider a fair coin has two sides and only one side is heads, and either side is likely to come up, then the theoretical probability of tossing heads is given by:

NOTE

Example of theoretical probability are found in determining the probability of drawing a certain card from a standard deck of cards. A standard deck has four suits: spades ( ), hearts (h ), diamonds () ), and clubs (k ). It has thirteen cards in each suit: ace, 2, 3, . . ., 10, jack, queen, and king. Each of these cards is equally likely to be drawn.

1).The probability of drawing a king is given by:

NOTE)

2).The probability of drawing a heart is given by:

3).The probability of drawing a face card (jack, queen, king) is given by:

These examples lead to four rules or facts about probability:• The probability of an event that cannot occur is 0.• The probability of an event that must occur is 1.• Every probability is a number between 0 and 1 inclusive.• The sum of the probabilities of all possible outcomes of an

experiment is 1.

Since any event will either occur or it will not occur, by rule 4 previously discussed, we get:

can also be stated as:

E.g.): So the probability of tossing a die and not rolling a 4 is:

Compound EventsA compound event is an event consisting of two or more simple events. Examples of simple events are: tossing a die and rolling a 5, picking a seven from a deck of cards, or flipping a coin and having a heads show up.

An example of a compound event is tossing a die and rolling a 5 or an even number. The notation for this kind of compound event is given by . This is the probability that event A or event B (or both) will occur.

So, out of the six numbers that can show up on top, we have four ways that we can roll either a 5 or an even number. The probability is given by:

Probability of rolling a 5 Probability of rolling an even number

Addition RuleThis leads to the Addition Rule for compound events. The statement of this rule is that the probability of either of two events occurring is the probability for the first event occurring plus the probability for the second event occurring minus the probability of both event occurring simultaneously.Stated mathematically the rule is given by:

E.g. the probability of drawing a 3 or a club from a standard deck of cards is:

Cards with a 3Cards with clubs

Card that is a 3 and a club

Multiplication Rule for Independent Events

• Independent events are events in which the occurrence of the events will not affect the probability of the occurrence of any of the other events. When we conduct two independent events we can determine the probability of a given outcome in the first event followed by another given outcome in the second event.

• If the set of crayons consists only of red, yellow, and blue, the probability of picking red is . The probability of tossing a die and rolling a 5 is .But the probability of picking red and rolling a 5 is given by:

The multiplication rule for independent events can be stated as: This rule can be extended for more than two independent events:

Multiplication Rule for Dependent Events

Dependent events are events that are not independent. The occurrence of one event affects the probability of the occurrence of other events. An example of dependent events is picking a card from a standard deck then picking another card from the remaining cards in the deck.

If A and B are the two events, we can express the probability that B will occur if A has already occurred by using the notation: This notation is generally read as “the probability of B, given A.

The multiplication rule can now be expanded to include dependent events. The rule now reads:

Of course, if A and B are independent, then:

Bayesian approach

• Bayes Theoremo Thomas Bayes

• Bayes Theorem

Binomial theorem• P(n,k,p)

o probability of k successes in n trialswhere the probability of success on any one trial is p

o “success” = some specific event or outcomeo k specified outcomeso n trialso p probability of the specified outcome in 1 trial

Binomial distribution

• Binomial theorem describes a theoretical distribution that can be plotted in two different ways:

o probability density function (PDF)

o cumulative density function (CDF)

The Standard Normal Distribution (Z)

All normal distributions can be converted into the standard normal curve by subtracting the mean and dividing by the standard deviation:

The Normal Distribution

X

f(X)

f

f

Changing μ shifts the distribution left or right.

Changing σ increases or decreases the spread.

Table

•Reference

• www.graves.k12.ky.us

• www.mente.elac.org

• www.docstoc.com

• www.stanford.edu

• www.chartwellyorke.com