What happens if you guess - Leigh Nataro

Prevodilac: Anja Saric Lektor: Mile Živković

Probability is an area of mathematics that is everywhere.

Verovatnoća je oblast matematike koja je prisutna svuda.

We hear about it in weather forecasts,

Čujemo o njoj u vremenskoj prognozi,

like there's an 80% chance of snow tomorrow.

na primer, verovatnoća da će sutra pasti sneg je 80%.

It's used in making predictions in sports,

Koristi se i kod pravljenja pretpostavki u sportu,

such as determining the odds for who will win the Super Bowl.

kao što je utvrđivanje šansi ko bi mogao da osvoji Super bol.

Probability is also used in helping to set auto insurance rates

Verovatnoća nam takođe pomaže pri postavljanju tarifa osiguranja kola

and it's what keeps casinos and lotteries in business.

i ona održava kockarnice i lutriju u životu.

How can probability affect you?

Kako verovatnoća može uticati na vas?

Let's look at a simple probability problem.

Pogledajmo jednostavan primer verovatnoće.

Does it pay to randomly guess on all 10 questions

Da li se isplati nasumično pogađati svih 10 pitanja

on a true/ false quiz?

testa na tačno/netačno?

In other words, if you were to toss a fair coin

Drugim rečima, ako bismo bacali novčić 10 puta

10 times, and use it to choose the answers,

i koristili ga pri odgovaranju na pitanja,

what is the probability you would get a perfect score?

koja je verovatnoća da ćemo dati sve tačne odgovore?

It seems simple enough. There are only two possible outcomes for each question.

Izgleda veoma jednostavno. Postoje samo dva moguća ishoda za svako pitanje.

But with a 10-question true/ false quiz,

Ali, sa testom od 10 pitanja tačno/netačno

postoji mnogo različitih načina da se iskombinuje tačno i netačno.

there are lots of possible ways to write down different combinations

of Ts and Fs. To understand how many different combinations,

Da bismo razumeli koliko je to raličitih kombinacija,

let's think about a much smaller true/ false quiz

hajde da uzmemo mnogo manji test tačno/netačno

with only two questions. You could answer

sa samo dva pitanja. Možete odgovoriti "tačno tačno",

"true true," or "false false," or one of each.

ili "netačno netačno", ili po jedno od oba.

First "false" then "true," or first "true" then "false."

Prvo "netačno" pa "tačno", ili prvo "tačno" pa "netačno",

So that's four different ways to write the answers for a two-question quiz.

Dakle to je četiri različita načina da se odgovori na test od dva pitanja.

What about a 10-question quiz?

Šta je sa testom od 10 pitanja?

Well, this time, there are too many to count and list by hand.

Pa, ovog puta imamo previše opcija da bismo ih brojali pešice.

In order to answer this question, we need to know the fundamental counting principle.

Da bismo odgovorili na ovo pitanje, treba znati osnovni princip brojanja.

The fundamental counting principle states

Osnovni princip brojanja kaže

that if there are A possible outcomes for one event,

da ako postoji A mogućih ishoda jednog događaja,

and B possible outcomes for another event,

i B mogućih ishoda drugog događaja,

then there are A times B ways to pair the outcomes.

onda postoji A puta B načina sparivanja ovih ishoda.

Clearly this works for a two-question true/ false quiz.

Jasno, ovo funkcioniše kod testa od dva tačno/netačno pitanja.

There are two different answers you could write for the first question,

Postoje dva različita odgovora koja možete dati na prvo pitanje

and two different answers you could write for the second question.

i dva različita odgovora koje možete dati na drugo pitanje.

That makes 2 times 2, or, 4 different ways to write the answers for a two-question quiz.

To znači 2 puta 2, ili 4 različita načina da odgovorite na test od dva pitanja.

Now let's consider the 10-question quiz.

Hajde sada da vidimo test od 10 pitanja.

To do this, we just need to extend the fundamental counting principle a bit.

Da bismo to uradili, moramo samo malo da produžimo osnovni princip brojanja.

We need to realize that there are two possible answers for each of the 10 questions.

Moramo razumeti da postoje dva moguća odgovora za svako od 10 pitanja.

So the number of possible outcomes is

Tako da je broj mogućih ishoda

2, times 2, times 2, times 2, times 2, times 2,

2, puta 2, puta 2, puta 2, puta 2, puta 2,

times 2, times 2, times 2, times 2.

puta 2, puta 2, puta 2, puta 2.

Or, a shorter way to say that is 2 to the 10th power,

Ili , ukratko rečeno, to je 2 na deseti,

which is equal to 1,024.

što je jednako 1024.

To znači da od svih 1024 načina da date odgovore na tačno/netačno testu

That means of all the ways you could write down your Ts and Fs,

only one of the 1,024 ways would match the teacher's answer key perfectly.

samo će jedan biti identičan rešenjima predavača.

So the probability of you getting a perfect score by guessing

Dakle, verovatnoća da ćete dati sve tačne odgovore pogađanjem

is only 1 out of 1,024,

je samo 1 u 1024,

or about a 10th of a percent.

ili oko desetine jednog procenta.

Clearly, guessing isn't a good idea.

Jasno je da pogađanje nije dobra ideja.

In fact, what would be the most common score

U stvari, koji bi bio najčešći rezultat

if you and all your friends were to always randomly guess

ako biste vi i svi vaši prijatelji nasumično pogađali odgovor

at every question on a 10-question true/ false quiz?

na svako od 10 pitanja tačo/netačno testa?

Well, not everyone would get exactly 5 out of 10.

Pa, ne biste svi imali 5 tačnih odgovora od 10.

But the average score, in the long run,

Ali, prosečan rezultat bi na duge staze

would be 5.

bio 5.

In a situation like this, there are two possible outcomes:

U situaciji kao što je ova, postoje dva moguća ishoda:

a question is right or wrong,

pitanje je tačno ili netačno,

and the probability of being right by guessing

i verovatnoća da ćete biti u pravu ako pogađate

is always the same: 1/2.

je uvek ista: 1/2.

To find the average number you would get right by guessing,

Da biste našli prosečan broj odgovora koje biste pogodili,

you multiply the number of questions

pomnožite broj pitanja

by the probability of getting the question right.

sa verovatnoćom da ćete dati tačan odgovor.

Here, that is 10 times 1/2, or 5.

U ovom slučaju, to je 10 puta 1/2, ili 5.

Hopefully you study for quizzes,

Nadam se da učite za testove,

jer se očigledno ne isplati pogađati.

since it clearly doesn't pay to guess.

Ali, jednom ste verovatno radili standardizovani test kao sto je SAT

But at one point, you probably took a standardized test like the SAT,

and most people have to guess on a few questions.

i većina ljudi mora da pogađa kod nekoliko pitanja.

If there are 20 questions and five possible answers

Ako postoji 20 pitanja i 5 mogućih odgovora na svako,

for each question, what is the probability you would get all 20 right

koja je verovatnoća da ćete pogoditi svih 20

by randomly guessing?

ako nasumično odgovarate?

And what should you expect your score to be?

I kakav rezultat biste mogli da očekujete?

Let's use the ideas from before.

Hajde da iskoristimo pređašnje ideje.

Prvo, pošto je verovatnoća da ćete dati tačan odgovor pogađanjem 1/5

First, since the probability of getting a question right by guessing is 1/5,

we would expect to get 1/5 of the 20 questions right.

očekivali biste da imate 1/5 od 20 pitanja tačno.

Yikes - that's only four questions!

Ups - to je samo četiri pitanja!

Are you thinking that the probability of getting all 20 questions correct is pretty small?

Da li mislite da je verovatnoća da imate svih 20 tačnih odgovora veoma mala?

Let's find out just how small.

Hajde da vidimo koliko je mala.

Do you recall the fundamental counting principle that was stated before?

Sećate li se osnovnog principa brojanja od malopre?

With five possible outcomes for each question,

Sa 5 mogućih ishoda za svako pitanje,

we would multiply 5 times 5 times 5 times 5 times...

pomnožili bismo 5 puta 5 puta 5 puta 5 puta...

Well, we would just use 5 as a factor

Pa, uzećemo samo 5 kao faktor

20 times, and 5 to the 20th power

20 puta, i 5 na dvadeseti

is 95 trillion, 365 billion, 431 million,

je 95 triliona, 365 milijardi, 431 milion,

648 thousand, 625. Wow - that's huge!

648 hiljada, 625. Vau - to je ogromno!

So the probability of getting all questions correct by randomly guessing

Dakle, verovatnoća da ćete imati sve tačne odgovore pogađanjem

je otprilike 1 u 95 triliona.

is about 1 in 95 trillion.