# 0.12 Probability  (Page 6/8)

 Page 6 / 8

We end the section by solving a problem called the Birthday Problem .

If there are 25 people in a room, what is the probability that at least two people have the same birthday?

Let event $E$ represent that at least two people have the same birthday.

We first find the probability that no two people have the same birthday.

We analyze as follows.

Suppose there are 365 days to every year. According to the multiplication axiom, there are ${\text{365}}^{\text{25}}$ possible birthdays for 25 people. Therefore, the sample space has ${\text{365}}^{\text{25}}$ elements. We are interested in the probability that no two people have the same birthday. There are 365 possible choices for the first person and since the second person must have a different birthday, there are 364 choices for the second, 363 for the third, and so on. Therefore,

$P\left(\text{No two have the same birthday}\right)=\frac{\text{365}\cdot \text{364}\cdot \text{363}\cdots \text{341}}{{\text{365}}^{\text{25}}}=\frac{\text{365}P\text{25}}{{\text{365}}^{\text{25}}}$

Since $P\left(\text{at least two people have the same birthday}\right)=1-P\left(\text{No two have the same birthday}\right),$

$P\left(\text{at least two people have the same birthday}\right)=1-\frac{\text{365}P\text{25}}{{\text{365}}^{\text{25}}}=\text{.}\text{5687}$

## Conditional probability

Suppose you and a friend wish to play a game that involves choosing a single card from a well-shuffled deck. Your friend deals you one card, face down, from the deck and offers you the following deal: If the card is a king, he will pay you $5, otherwise, you pay him$1. Should you play the game?

You reason in the following manner. Since there are four kings in the deck, the probability of obtaining a king is $4/\text{52}$ or $1/\text{13}$ . And, probability of not obtaining a king is $\text{12}/\text{13}$ . This implies that the ratio of your winning to losing is 1 to 12, while the payoff ratio is only $1 to$5. Therefore, you determine that you should not play.

Now consider the following scenario. While your friend was dealing the card, you happened to get a glance of it and noticed that the card was a face card. Should you, now, play the game?

Since there are 12 face cards in the deck, the total elements in the sample space are no longer 52, but just 12. This means the chance of obtaining a king is $4/\text{12}$ or $1/3$ . So your chance of winning is $1/3$ and of losing $2/3$ . This makes your winning to losing ratio 1 to 2 which fares much better with the payoff ratio of $1 to$5. This time, you determine that you should play.

In the second part of the above example, we were finding the probability of obtaining a king knowing that a face card had shown. This is an example of conditional probability . Whenever we are finding the probability of an event E under the condition that another event F has happened, we are finding conditional probability.

The symbol $P\left(E\mid F\right)$ denotes the problem of finding the probability of $E$ given that $F$ has occurred. We read $P\left(E\mid F\right)$ as "the probability of $E$ , given $F$ ."

A family has three children. Find the conditional probability of having two boys and a girl given that the first born is a boy.

Let event $E$ be that the family has two boys and a girl, and $F$ the event that the first born is a boy.

First, we list the sample space for a family of three children as follows.

$S=\left\{\text{BBB},\text{BBG},\text{BGB},\text{BGG},\text{GBB},\text{GBG},\text{GGB},\text{GGG}\right\}$

Since we know that the first born is a boy, our possibilities narrow down to four outcomes, $\text{BBB}$ , $\text{BBG}$ , $\text{BGB}$ , and $\text{BGG}$ .

Among the four, $\text{BBG}$ and $\text{BGB}$ represent two boys and a girl.

Therefore, $\begin{array}{}P\left(E\mid F\\ =2/4\end{array}$ or $1/2$ .

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
If March sales will be up from February by 10%, 15%, and 20% at Place I, Place II, and Place III, respectively, find the expected number of hot dogs, and corn dogs to be sold
8. It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges$12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost?
At De Anza College, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?