Lesson: Rewriting Questions

Comment on Rewriting Questions

Another possible solution to the question might be that the probability that the product is positive is equal to 1 - the probability that the product is negative, which is 1 - 12/42 or 2/7 and with the answer is 5/7. Where do I fail, because in the video you said that the answer is 3/7? This is the second time when using a different approach, which should be correct I receive different answer.
greenlight-admin's picture

P(product is negative) = P(1st # is neg and 2nd # is pos OR 1st # is pos and 2nd # is neg)
= P(1st # is neg and 2nd # is pos) + P(1st # is pos and 2nd # is neg)
= (4/7 x 3/6) + (3/7)(4/6)
= 12/42 + 12/42
= 24/42
= 4/7

So, P(product is positive) = 1 - 4/7
= 3/7

Thank you for summarizing at the end all strategies and telling us when to use them. Your way of teaching is very effective.

Another way to solve the problem is to directly find the probability that the product of the first and second number selected are positive, as follows:
P(prod pos) = P(#1 neg and #2 neg) + P(#1 pos and #2 pos)
P(#1 neg and #2 neg) = P(#1 neg) * P(#2 pos | #1 neg)
= 4/7(3/6) = 12/42

P(#1 pos and #2 pos) = P(#1 pos) * P(#2 pos | #1 pos)
= 3/7 * (2/7) = 6/42

P(prod pos) = 12/42 + 6/42 = 18/42 = 3/7
greenlight-admin's picture

Perfect!!!!

Hello Brent,
I always have this doubt in probability- It is a confusion between Mutually exclusive events and Independent events. Though I understand both these concepts individually, but I get bit confused when it comes to questions. Like here in this question explained in module, how did you tell that the events of selecting 1 boy and 2 boys are mutually exclusive? Please explain. Also will mutually exclusive events be independent events as well? Please clarify. Pls explain the other method of solving also, that is the first method of : P(atleast 1 boy) = P(1 boy) + P(2 boys). How to calculate P(2 boys) here? I have doubt. Will it be 2/8? which is 1/4?
greenlight-admin's picture

QUESTION: How did you tell that the events of selecting 1 boy and 2 boys are mutually exclusive?

If events A and B are mutually exclusive, then P(A AND B) = 0.

In this question, the probability of selecting exactly 1 boy AND selecting exactly 2 boys is ZERO. That is, if you select 2 people, it cannot be the case that those two people consist of 1 boy AND consist of 2 boys.

QUESTION: Will mutually exclusive events be independent events as well?

If events A and B are possible (that is P(A) > 0 and P(B) > 0), then the two events cannot be both independent and mutually exclusive.

QUESTION: Pls explain the other method of solving also, that is the first method of : P(at least 1 boy) = P(1 boy) + P(2 boys). How to calculate P(2 boys) here?

We have a few ways to calculate both parts. Here's one option.

P(1 boy) = P(1st person is boy and 2nd person is girl OR 1st person is girl and 2nd person is boy)

= [P(1st person is boy) x P(2nd person is girl)] + [P(1st person is girl) x P(2nd person is boy)]

= [4/8 x 4/7] + [4/8 x 4/7]

= 2/7 + 2/7

= 4/7

P(2 boys) = P(1st person is boy AND 2nd person is boy)

= P(1st person is boy) x P(2nd person is boy)

= 4/8 x 3/7

= 3/14

So, P(at least 1 boy) = P(1 boy) + P(2 boys)
= 4/7 + 3/14
= 11/14

Thanks a ton Brent!

Hi Brent,

I did it this way:

the counter probability = p(one number is positive) * P(one number is negetive) = (4/7) * (1/2) = 2/7

Hence, the probability is 1 - 2/7 = 3/7.
greenlight-admin's picture

I'm not sure about that solution.
Notice that 1 - 2/7 = 5/7 (not 3/7)

Hi Brent,

Can you solve this question using formula please
You're playing a game where you defend your village from an orc invasion. There are 3 characters (elf, hobbit, or human) and 5 defense tools (magic, sword, shield, slingshot, or umbrella) to pick from.
If you randomly choose your character and tool, what is the probability that you won't be a hobbit or use an umbrella?

My solution

P[(Wont be using hobbit OR Wont be using umbrella)] = P(Wont be using hobbit) + P(Wont be using umbrella) - P(Wont be using Hobbit AND Wont be using umbrella).

2/15 + 4/15 - 1/15 = 1/3 which is wrong. answer is 8/15.
Please help. Thank you in advance.
greenlight-admin's picture

This is actually an AND probability situation, since we want a non-hobbit AND we want a non-umbrella.

When we choose an character, P(selection is NOT an hobbit) = 2/3
When we choose a weapon, P(selection is NOT an umbrella) = 4/5

The two events are independent.
So, P(selection is a non-hobbit AND selection is non-umbrella) = P(character selection is a non-hobbit) x P(weapon selection is a non-umbrella)
= 2/3 x 4/5
= 8/15

Cheers,
Brent

So I just read the word OR and went for it, What can I do in future to identify if they are in OR or AND category ?. Cause here they mentioned OR in the question but they expected us to apply AND formula.
greenlight-admin's picture

The best advice I can give you is to try to reword the question so that you eliminate the NOT/WON'T language (e.g., Won't be a Hobbit and Won't be an umbrella).
That should help quite a bit.

Hi Brent,

I have tried solving the second question using the complement, can you please check my logic? Thanks

P(selecting AT LEAST 1 boy) = 1 - P(selecting zero boys)

Now we calculate the prob of selecting zero boys:
P(selecting zero boys) = P (select one girl AND select another girl) =
P(G1) + P(G2|G1) =
= P(G1) + P(G2|G1) = 4/8*3/7 = 12/56

Thus the probability of selecting AT LEAST one boy is 1-12/56 = 44/56

Is this ok? Thanks a lot!
greenlight-admin's picture

Great work!
That's exactly how I'd solve it.

can you please explain what would be P(G2|G1)
greenlight-admin's picture

The vertical line stands for "given that"

P(G2|G1) represents the probability that the second selected student is a girl (G2) given that the first selected student is a girl (G1)

Cheers,
Brent

I totally understand the concept Mr. Bernt. I need to understand the application in this particular problem. Numerical form of this particular problem. 4/8+3/7? How do we achieve 11/14 applying the formula shown in the video.
Looking forward thanks
greenlight-admin's picture

With that particular solution, we're subtracting the PRODUCT (4/8)(3/7) from 1.
In other words: P(at least 1 boy) = 1 - P(both selections are girls)
= 1 - (4/8)(3/7)
= 1 - (1/2)(3/7)
= 1 - 3/14
= 11/14

Does that help?

I'm scratching my head on this Q: https://greprepclub.com/forum/topic8583.html

Not sure how this conclusion is met: "First of all, it's useful to recognize that P(2nd ball is NOT red and the 3rd ball is yellow) is the SAME as P(1st ball is NOT red and the 2nd ball is yellow)"
greenlight-admin's picture

Great question!
Here's my response: https://greprepclub.com/forum/topic8583.html#p42572

Cheers,
Brent

Hi Brent,
In case, we need to find P(product is neg).Is the value of P(Product is negative ) is 4/7?? Please correct me.. Thanks a ton.
greenlight-admin's picture

That's correct.

In that question, we learned that P(product is positive) = 3/7.
Since it is impossible to get a product equal to zero, we can conclude that the product must be either positive or negative.

So, if P(product is positive) = 3/7, then P(product is negative)

Cheers,
Brent

I personally found all that rewriting for the integer problem way too taxing. What I did was find the number of pairs that can be formed from those 7 number using combinations, so 7C2, which is 21. That's the denominator. Then I just picked out all the pairs of numbers that make a positive product. I found 9 pairs. 9/21 = 3/7. Boom.
greenlight-admin's picture

That works!

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