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Chapter 3-3. High Difficulty Probability Problems

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1. Problem 1

2. Problem 2

3. Problem 3

4. Problem 4

1. Problem 1

⑴ Problem

① A : A is a random variable representing the number of times a heads appears when tossing a coin n+1 times.

② B : B is a random variable representing the number of times a heads appears when tossing a coin n times.

③ P(A ＞ B)?

⑵ Solution

① A* : A* is a random variable representing the number of times a heads appears when tossing a coin n times.

② B* : B* is a random variable representing the number of times a heads appears when tossing a coin n times.

③ P(A* ＞ B) = P(A ＜ B*)

④ P(A* ＞ B) + P(A = B) + P(A ＜ B*) = 1

⑤ P(A ＞ B) = P(A ＞ B | A* ＞ B) + P(A ＞ B | A = B) + P(A ＞ B | A ＜ B*)

○ P(A ＞ B ∩ A* ＞ B*) = 1

○ P(A ＞ B ∩ A* = B*) = 1/2

○ P(A ＜ B ∩ A* ＜ B*) = 0

⑥ P(A ＞ B) = P(A* ＞ B) + 1/2 P(A = B) = 1/2 (P(A ＞ B) + P(A* = B) + P(A ＜ B*)) = 1/2

⑶ Comment

① Calculating algebraically is challenging

2. Problem 2. Bayes’ Theorem

⑴ Premises

① A pharmaceutical company claims that their new drug improves memory with a probability of 0.8.

② A memory test is conducted on 20 subjects.

③ If the drug is effective, the test scores always increase.

④ If the drug is not effective, there is a 0.5 probability of test scores increasing.

⑤ If the drug is not effective, there is a 0.5 probability of test scores decreasing.

⑥ Among the 20 subjects, 16 show an increase in test scores.

⑵ Definitions

① M : Event of memory improvement

② N : Event of no change in memory

③ S : Event of an increase in test scores

④ T : Event of 16 out of 20 subjects showing an increase in test scores

⑤ F₁ : Event of the pharmaceutical company’s claim being true

⑥ F₂ : Event of the pharmaceutical company’s claim being false

⑶ Problem 1. P(T | F₁) : Probability of the result occurring if the pharmaceutical company’s claim is true

① Approach 1.

② Approach 2.

⑷ Problem 2. P(T | F₂) : Probability of the result occurring if the pharmaceutical company’s claim is false

⑸ Problem 3. P(F₁ | T) : Probability that the pharmaceutical company’s claim is true after conducting the test

⑹ Problem 4. P(M | T) : Probability of memory improvement after conducting the test

3. Problem 3. Joint Probability Distribution

⑴ Problem

① There is a beach with a length of ℓ km starting from the origin.

② Select an arbitrary point.

③ Select an arbitrary point within the interval from the origin to the selected point.

④ Y : Distance from the origin to the initially selected point

⑤ X : Distance from the origin to the finally selected point

⑵ p(x)

4. Problem 4

⑴ Problem

① X : Number of people who correctly receive their own hats after randomly redistributing hats from n people.

② Find E(X) and VAR(X).

⑵ Change of Perspective

① X_i : If the i-th person receives their own hat, X_i = 1; otherwise, X_i = 0.

② X = X₁ + ··· + X_n

⑶ E(X)

① Key : E(X_i) = 1/n

② Interpretation 1: Approach based on counting possibilities

③ Interpretation 2: The expected value of whether the i-th person receives their own hat or not is constant due to symmetry.

⑷ VAR(X)

Input : 2019.07.04 10:13