Lesson: Guessing Strategies

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Hi Brent!

For a certain probability experiment, the probability that event A will occur is 1/2 and the probability that event B will occur is 1/3. Which of the following values could be the probability that the event AUB (that is, the event A or B, or both) will occur?

indicate all such values
a)1/3 b)1/2 c)3/4

for P(A or B) i did like P(A)*P(~A) + P(B)*P(~B)
1/2 * 2/3 + 1/3 * 1/2 for that i got 3/4,

But in ETS quant guide b) and c) both are answers.
Please let me know about P(AUB) (A union B) and how b) is also answer.
you have explained about P(A intersection B), both A and B can occur, in mutually exclusive function.
greenlight-admin's picture

We want P(A or B)

P(A or B) = P(A) + P(B) - P(A and B)

= 1/2 + 1/3 - P(A and B)

= 5/6 - P(A and B)

This means P(A or B) must be less than or equal to 5/6. At this point, it comes down to the value of P(A and B)
However, we can be certain that P(A or B) cannot be less than P(A), and P(A and B) cannot be less than P(B). How do we know this?

Well, if P(A) = 1/2, then it cannot be the case that P(A OR B) is less than 1/2.

In other words, P(A or B) cannot be less than 1/2, and P(A or B) cannot be less than 1/3

Since 1/3 < 1/2, we can conclude that P(A or B) cannot be less than 1/2

So, P(A or B) must be less than or equal to 5/6 AND greater than or equal to 1/2

This allows us to ELIMINATE answer choice A, and we are left with answer choices B and C

ASIDE: Here's an experiment in which P(A or B) = 1/2

Select ONE number from the set: {1, 2, 3, 10, 15, 20}
Event A: The selected number is a multiple of 5
Event B: The selected number is a multiple of 10

Here, P(A) = 3/6 = 1/2 and P(B) = 2/6 = 1/3
Also, P(A and B) = 2/6 = 1/3

So, P(A or B) = P(A) + P(B) - P(A and B)
= 1/2 + 1/3 - 1/3
= 1/2

Does that help?

" Well, if P(A) = 1/2, then it cannot be the case that P(A OR B) is less than 1/2.
In other words, P(A or B) cannot be less than 1/2, and P(A or B) cannot be less than 1/3
Since 1/3 < 1/2, we can conclude that P(A or B) cannot be less than 1/2
So, P(A or B) must be less than or equal to 5/6 AND greater than or equal to 1/2 "

above what you explained was crucial to solve this, but i am quite not sure how P(A or B) can not be less than P(A) 1/2 and P(B) 1/3.
greenlight-admin's picture

If I add another possible event to the probability, then the probability of one event OR another event cannot be less than the probability of one even alone. Here's an example:

Let's say the probability is 0.5 that it rains tomorrow, and let's also say the probability is 0.3 that my dog eats a bee tomorrow.

What is the probability that it rains tomorrow OR my dog eats a bee tomorrow OR both? That is P(rains OR dog eats bee tomorrow)

Well, P(rains tomorrow) = 0.5

No matter what, there's a 50% likelihood that it rains tomorrow. So, adding a second possible event (e.g., dog eating a bee) must be AT LEAST AS likely as it raining tomorrow.

That is, P(rains OR dog eats bee tomorrow) must be greater than or equal to P(rain tomorrow).

Put another way: Let's say I'll give you one million dollars if a certain event happens. Furthermore, I'm going to give you three cases to choose from:

case a: It rains tomorrow
case b: My dog eats a bee tomorrow
case c: It rains tomorrow OR my dog eats a bee tomorrow

Which of these cases is most likely to occur? Well, P(rain tomorrow) = 0.5. So, P(rain tomorrow OR dog eats a bee tomorrow cannot be less than 0.5

Does that help?

Oh wow!, I have understood now. Certainly you are one of the best teacher in my learning course.

Why have you said that answer choices 0.027,0.343 are greater than 0.5? I don't get it.
greenlight-admin's picture

Hi Deepak,

That's not quite what I say in the video. I ask "Do you feel that the probability is greater than or less than 0.5?"

If you feel that the probability is GREATER THAN 0.5, then we can ELIMINATE answer choices A and B since those values are less than 0.5.

Cheers,
Brent

I want to be member
greenlight-admin's picture

Great - here's where to start: https://www.greenlighttestprep.com/prices

Cheers,
Brent

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