Lesson: Introduction to Probability

Comment on Introduction to Probability

in the set of {3,5,9,12,15,18,21} 5 is neither prime nor even.Thus the probability should be 3/7

Oh my bad. 5 is a prime number. sorry :| i got confused

For the linked question: http://greprepclub.com/forum/gre-math-challenge-15-if-one-number-is-chosen-at-random-334.html

Given that any multiple of 8 will also be a multiple of 2,
why can't we just divide 1000/8 to find the number of multiples of 8, and make that (125) the numerator over the denominator of 1000?

greenlight-admin's picture

Yes, we can. In fact that's what I did in my solution.
However, my solution also includes a word of caution.

Notice what happens if the question were: If one number is chosen at random from the first 10 positive integers, then what is the probability that the number is a multiple of 2 and 8?

In this case, dividing 10 by 8 will not yield the correct answer. So, we need to be careful to not try to apply a generic approach to all similar questions.

With the last practice problem in the video (choosing Amir from a set of 5), how can I know when the problem involves combinations? I thought the denominator would be 5 because it's from a group of 5 people.
greenlight-admin's picture

If we were randomly selecting ONE person from the group of five people, then there would be 5 ways to do that:
1) A
2) B
3) C
4) D
5) E

We are randomly selecting TWO people from the group of five people. There are 10 ways to do that:
1) AB
2) AC
3) AD
4) AE
5) BC
6) BD
7) BE
8) CD
9) CE
10) DE
Another way to make this calculation is to use combinations, since the order in which we select the 2 people does not matter. For example, selecting A then B (AB) is the same as selecting B then A (BA)

More here: https://www.greenlighttestprep.com/module/gre-counting/video/787

https://magoosh.com/gre/2012/gre-probability-practice-question-of-the-week-33/

The solution for the problem is not mentioned, or at least I cant view it. My understanding is to build a 90 degree triangle inside the circle with legs 2 each and then the hypotenuse is going to be 2 root 2. After that my mind goes numb. Can you please explain this.

Tossing a fair coin, the probability of getting 'heads' is 0.5. If three coins are tossed simultaneously, what is the probability of getting at least 2 heads?
greenlight-admin's picture

Given that there aren't many possible outcomes when we flip 3 coins, a relatively fast and painless solution is to list all possible outcomes.

We'll list the outcomes in terms of H's and T's, where the 1st letter represents the 1st coin, the 2nd letter represents the 2nd coin, etc.

The outcomes are as follows:
1) HHH
2) HHT
3) HTH
4) THH
5) TTH
6) THT
7) HTT
8) TTT

Now check to see which outcomes have at least 2 heads.
They are outcomes 1, 2, 3 and 4

So, 4 of the 8 outcomes have at least 2 heads

P(at least 2 heads) = 4/8 = 1/2

Answer 1/2

Cheers,
Brent

Hi Brent, for the example given here, "5 people are chosen 2 at a time, chances of Amir being chosen?"
I looked at it this way:

There are 5 people to choose from, so my denominator is 5-----> (some number, aka numerator/5)
The chances of Amir being chosen the first time around out of 5 people is one time, therefore 1/5.
The chances of Amir being chosen again a second time, once again, 1/5.
Add up both fractions 1/5 + 1/5 =2/5, or .40

Is this a valid approach?\

Thanks much!
greenlight-admin's picture

GREAT QUESTION! TOUGH QUESTION!

I read your post last night and, since your strategy didn't "feel" right to me, I started looking for counter-examples (using the same strategy you describe) to show the flaw in your strategy . . . but your strategy worked every time!

So, I decided to sleep on it.

Now that I've had plenty of time to think about it, I can tell you that the main issue with your solution is that it doesn't adhere to the general probability formula.
That is, P(Event A occurs) = (# of ways event A can occur)/(total number of possible outcomes)

For example, in your solution, you say "There are 5 people to choose from, so the denominator is 5"
However, what does the 5 represent?
It doesn't represent the total number of possible outcomes, since there are 10 ways in which we can choose TWO people from 5 people.
The same can be said of the numbers you're using in the numerator; they don't represent values needed to apply the general probability formula.

HOWEVER (and it's a big HOWEVER), even though the numbers you're using don't represent values needed to apply the general probability formula, those values do, indeed, work.
In fact, we can PROVE they work.

Based on your strategy, we can write: IF JOE IS AMONG N PEOPLE, AND 2 PEOPLE ARE RANDOMLY SELECTED, THEN P(JOE IS SELECTED) = 2/N

Here's the proof:
If there are N people, then we can select 2 people in NC2 (N choose 2) different ways.
NC2 = (N)(N-1)/2
So, this is our denominator

In how many of the (N)(N-1)/2 possible outcomes is Joe chosen?
Well, once we make Joe one of the selected people, there are N-1 people remaining to join Joe.
So, there are N-1 outcomes in which Joe is one of the selected people.
This is our numerator

So, P(Joe is selected) = (N-1)/[(N)(N-1)/2] = (N-1)[2/(N)(N-1)] = 2/N
Voila!!

We can even extend this property to say:
IF JOE IS AMONG N PEOPLE, AND 3 PEOPLE ARE RANDOMLY SELECTED, THEN P(JOE IS SELECTED) = 3/N

We can also say:
IF JOE IS AMONG N PEOPLE, AND 4 PEOPLE ARE RANDOMLY SELECTED, THEN P(JOE IS SELECTED) = 4/N

In general, we can say:
IF JOE IS AMONG N PEOPLE, AND K PEOPLE ARE RANDOMLY SELECTED, THEN P(JOE IS SELECTED) = K/N

Cheers,
Brent

Hi Brent,
Could you explain more the last two lines on your solution of question number 11 from Reinforcement Activities
greenlight-admin's picture

Hi Omer,
In the future, please include the link of the question so I'm certain which question you're referring to.
My previous solution needed a little work in clarifying matters.
I have added a few more lines at the ends to help.
Please let me know what you think: https://greprepclub.com/forum/if-an-integer-greater-than-100-and-less-th...

Cheers,
Brent

Thanks a million, dear Brent. I will do that.

Hi Brent ,
For https://greprepclub.com/forum/if-3-different-integers-are-randomly-selected-from-the-integ-13816.html

I didnt get this line

"When we check the answer choices, we see that only one answer choice (C) has a denominator that's a FACTOR of 440."

could you please help me how did you approach to option c

Thanks
greenlight-admin's picture

Question link: https://greprepclub.com/forum/if-3-different-integers-are-randomly-selec...

When we have a fraction that can be reduced to a simpler fraction, the denominator of the simplified fraction will always be a factor of the original fraction.

To see what I mean, let's simplify some fractions.
3/6 = 1/2. Notice that 2 is a factor of 6
15/20 = 3/4. Notice that 4 is a factor of 10
28/100 = 7/25. Notice that 25 is a factor of 100
12/440 = 3/110. Notice that 110 is a factor of 440
77/440 = 7/40. Notice that 40 is a factor of 440

So, if we simplify x/440, we know that the denominator of the simplified fraction must be a factor of 440.
Does that help?

Hi Brent,

for this question: https://greprepclub.com/forum/one-person-is-to-be-selected-at-random-from-a-group-of-25-pe-8094.html

Isn't the number of males are 11 out of 25, as such should't be the number of males that are born before 1960 be: 0.28 x 11 (instead 0.28 x 25, since we know that among the total of 25, there are a non-male people).
greenlight-admin's picture

Question link: https://greprepclub.com/forum/one-person-is-to-be-selected-at-random-fro...

Great question!
It all comes down to the wording.
That is, it all comes down to the group of people from which we are selecting one person.

The question tells us that "the probability that the selected person will be a male who was born before 1960 is 0.28."
We're selecting 1 person out of all 25 people (not just 1 person from the 11 males)
So, if B equals the number of males born before 1960, then B/25 = 0.28

If the question had told us that we're randomly selecting one of the males, and that the probability that the selected person will be a male who was born before 1960 is 0.28, then we'd write: B/11 = 0.28

Does that help?

Cheers,
Brent

Thank you Brent, yes it does!

Hi Brent! Question on this one: https://greprepclub.com/forum/gre-math-challenge-15-if-one-number-is-chosen-at-random-334.html

Why can't we use the p(A and B) = p(A) X p(B) rule for this?

P(divisible by 2) is 1/2 while P(divisible by 8) is 1/8, can we multiply the two to get 1/16th as the answer?
greenlight-admin's picture

Question link: https://greprepclub.com/forum/gre-math-challenge-15-if-one-number-is-cho...
The question: If one number is chosen at random from the first 1000 positive integers,then what is the probability that the number is a multiple of 2 and 8?

We can definitely use probability rules to solve this question (for students just starting probability, this particular concept is covered in the following video: https://www.greenlighttestprep.com/module/gre-probability/video/752)

p(A and B) = p(A) X p(B) when the two events are independent. In this case the two events are not independent.

That said we can always use the formula p(A and B) = p(A) X p(B|A)

So, P(# is a multiple of 2 AND 8) = P(# is multiple of 2) x P(# is multiple of 8 | # is multiple of 2)
= 1/2 x 1/4
= 1/8

Let's take a closer look at P(# is multiple of 8 | # is multiple of 2)
In other words: P(# is multiple of 8 GIVEN THAT # is multiple of 2)
If the number is a multiple of 2, then the number can be 2, 4, 6, 8, 10, 12, 14, 16, 18,...998, 1000
So, GIVEN THAT the number is in the set 2, 4, 6, 8, 10, 12, 14, 16, 18,...998, 1000, what is the probability that the number is divisible by 8?
Well, as you can see, 1 out of every 4 numbers is divisible by 8.
So, P(# is multiple of 8 | # is multiple of 2) = 1/4

Does that help?

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