# Help with my 2 easy but hard probability questions

• sneaky666
By definition, the probability of an event A is the measure of the set A, denoted as P(A), relative to the sample space. This means that P(A) is a number between 0 and 1, inclusive.For any two events A and B, P(A∪B) = P(A) + P(B) - P(A∩B), where A∪B represents the union of A and B, and A∩B represents the intersection of A and B.For any event A, P(A^c) = 1 - P(A), where A^c represents the complement of A.For any two events A and B, if A⊂B, then P(A) ≤
sneaky666
1.
Suppose P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
a) Must we have P((0,1/2]) <= 1/3 ?
b) Must we have P([0,1/2]) <= 1/3 ?

a)

P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
lim P([0,1/2])
n-> infinity

so then P((0,1/2]) is a subset of P([1/n,1/2]) for all n = 1,2,3,...
because any number in (0,1/2] can lie in [1/n,1/2] provided the correct n value is in -> 1,2,3...

so true
b)

P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
lim P([0,1/2])
n-> infinity

so then P([0,1/2]) does not equal P([1/n,1/2]) for all n = 1,2,3,...
because as n approaches infinity, the limit of 1/n goes to 0, but doesn't touch 0

so false

2.

Suppose P((0,1/2]) = 1/3. Prove that there is some n such that P([1/n,1/2]) > 1/4.

lim P([1/n,1/2]) = P([0,1/2]) > 1/3 > 1/4
n -> infinity

i don't think this is entirely right, i sort of guessed...

sneaky666 said:
1.
Suppose P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
a) Must we have P((0,1/2]) <= 1/3 ?
b) Must we have P([0,1/2]) <= 1/3 ?

a)

P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
lim P([0,1/2])
n-> infinity

so then P((0,1/2]) is a subset of P([1/n,1/2]) for all n = 1,2,3,...
because any number in (0,1/2] can lie in [1/n,1/2] provided the correct n value is in -> 1,2,3...

so true
P((0,1/2]) and P([1/n,1/2]) are numbers. One is not a subset of the other. Also, you have the relationship backwards: [1/n,1/2]⊂(0,1/2], not (0,1/2)⊂[1/n,1/2] as you claimed.
b)

P([1/n,1/2])<= 1/3 for all n = 1,2,3,...
lim P([0,1/2])
n-> infinity

so then P([0,1/2]) does not equal P([1/n,1/2]) for all n = 1,2,3,...
because as n approaches infinity, the limit of 1/n goes to 0, but doesn't touch 0

so false
Right.
2.

Suppose P((0,1/2]) = 1/3. Prove that there is some n such that P([1/n,1/2]) > 1/4.

lim P([1/n,1/2]) = P([0,1/2]) > 1/3 > 1/4
n -> infinity

i don't think this is entirely right, i sort of guessed...
It's not correct. For one thing, you don't know that P([0,1/2])>1/3.

What kind of properties about sets and set operations as it pertains to probabilities do you know? (This would have been what you should have written under "relevant equations" in the provided template.)

## 1. What are the two probability questions?

The two probability questions are typically described as "easy but hard" because they may seem simple at first glance, but require careful thinking and understanding of probability concepts to solve correctly.

## 2. What is the difference between easy and hard probability questions?

The difference between easy and hard probability questions lies in the level of complexity and the type of thinking required to solve them. Easy probability questions typically involve simple calculations and straightforward applications of probability rules, while hard probability questions may involve more advanced concepts and require creative problem-solving approaches.

## 3. How can I improve my probability problem-solving skills?

To improve your probability problem-solving skills, it is important to have a strong understanding of basic probability concepts and rules. Practice solving different types of probability problems and try to understand the underlying principles behind them. Additionally, studying real-world examples and applications of probability can help build your intuition and ability to solve more complex problems.

## 4. Can you provide tips for approaching probability questions?

When approaching probability questions, it can be helpful to break down the problem into smaller parts and identify what information is given and what is being asked. It is also important to carefully read and understand the question, as well as any given assumptions or conditions. Drawing diagrams or using visual aids can also aid in understanding and solving the problem.

## 5. Are there any common mistakes to avoid when solving probability problems?

One common mistake to avoid when solving probability problems is assuming that events are independent when they are actually dependent. It is important to carefully consider the given information and any assumptions or conditions stated in the problem. Additionally, it is important to check your calculations and make sure they are accurate, as small errors can lead to incorrect solutions.

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