Probability and Statistics - Quiz Old

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[ ] : A procedure we perform (quite often hypothetical) that produces some result. Often the letter $E$  is used to designate it.

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[ ] : A possible result of an experiment. The Greek letter $\xi$   is often used to represent it.

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[ ] : A certain set of outcomes of an experiment.

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[ ] (2 words) : the collection or set of "all possible" distinct (collectively exhaustive and mutually exclusive) outcomes of an experiment. The letter $S$  is used to designate it

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[ ] : a statement that is taken to be true, to serve as a premise or starting point for further reasoning and argument. "공리" in Korean.

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19! = ?

factorial(19) = ?

참고로 답은 18자리의 정수이다.

Multiple Choice

You are dealt 5 cards from a standard 52 card deck. What is the probability that you are dealt a “four of a kind” in ace ?

(52장의 카드에서 5장을 받았을 때 "에이스 포카"가 될 확률은?)

• 3.8476929233231754e-07

  (0.00000038476929..)

• 1.5390771693292702e-06

  (0.0000015390771...)

• 1.846892603195124e-05

  (0.0000184689...)

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On their way home from track practice, 8 high school freshmen stop at a restaurant, where each of them has one of the following: a cheese-burger, a hot dog, a taco, or a fish sandwich.

How many different purchases are possible?

​Quiz - Bayes Theorem

Some text here about the topic of discussion

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The [ ] Probability : $\Pr\left(A,B\right)\equiv\Pr\left(A\cap B\right)\$  

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[ ] : a statement that is taken to be true, to serve as a premise or starting point for further reasoning and argument. "공리" in Korean.

Poll

다양한 문서 작성/편집 S/W에서 쓰이는 Style/서식/Template 등의 기능에 대해 알고 있는가? 활용하는가?

• 알지 못함

• 알고는 있으나 거의 사용하지 않음

• 알고 있으며 사용함

Multiple Select

The Conditional Probability : $$\Pr\left(A\mid B\right)\ =$$  

(Be careful, there can be more than one correct answer. Full points are awarded only if all correct answers are selected)

• $$\frac{\Pr\left(A,B\right)}{\Pr\left(A\right)}$$  

• $$\frac{\Pr\left(A,B\right)}{\Pr\left(B\right)}$$  

• $$\frac{\Pr\left(A\cap B\right)}{\Pr\left(B\right)}$$  

• $$\frac{\Pr\left(A\cup B\right)}{\Pr\left(A\right)}$$  

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In a game of poker, you are dealt five cards from a standard 52 card deck.

What is the probability that you are dealt a flush in heart ?

(소수점 이하 6번째 자리까지 정확하게 적어라. ex) 3.141592)

Multiple Choice

Mr. B, Mr. C and Mr. D are land owners of a county called "U" and they are owning 1/2, 1/3 and 1/6 of the land of the county "U" respectively. Mr. A purchases 1/3 of the Mr. B's land, 1/6 of the Mr. C's land and 1/2 of the Mr. D's land.

How much, in percent, of the land of "U" is purchased by Mr. A?

• $$\frac{13}{36}\cong36.1\%$$

• $$\frac{1}{2}=50\%$$  

• $$\frac{11}{36}\cong30.6\%$$  

• $$\frac{1}{3}\cong33.3\%$$  

Multiple Choice

Mr. B, Mr. C and Mr. D are land owners of a county called "U" and they are owning 1/2, 1/3 and 1/6 of the land of the county "U" respectively. Mr. A purchases 1/3 of the Mr. B's land, 1/6 of the Mr. C's land and 1/2 of the Mr. D's land.

How much, in percent, of the land of "A" is originally owned by Mr. D?

• $$\frac{3}{10}=30\%$$

• $$\frac{1}{4}=25\%$$  

• $$\frac{11}{36}\cong30.6\%$$  

• $$\frac{3}{11}\cong27.3\%$$  

Multiple Choice

In the Auditorium Example in the lecture material,

Find the probability that Seat 15 was selected given that Row 20 was selected.

$$\Pr\left(S=15\mid R=20\right)=$$  

• $$\frac{1}{20}$$  

• $$\frac{1}{15}$$  

• $$\frac{1}{30}$$  

• $$\frac{1}{\left(11+12+\cdots+30\right)}$$  

Multiple Choice

In the Auditorium Example in the lecture material,

Find the probability that Seat 15 was selected.

$$\Pr\left(S=15\right)\approx$$  

• 0.0271

• 0.0342 

• 0.0087 

• 0.175 

Multiple Choice

In the Auditorium Example in the lecture material,

Find the probability that Row 20 was selected given that Seat 15 was selected.

$$\Pr\left(R=20\mid S=15\right)\approx$$  

• 0.324

• 0.0313

• 0.1324

• 0.175 

Multiple Choice

Medical Test Example

You’ve got some medical tests. The results turned out that you tested positive for a very rare disease that is known to affect about 0.1% of the population. It is a nasty disease with horrible consequences.

You asked your doctor “How certain is it that I have this disease ?”. The doctor says that the test correctly identify 99% of people that have the disease and only incorrectly identify 1% of people who don’t have the disease.

What are the chances that you actually have this disease ?

• 0.701 (70.1%)

• 0.168 (16.8%)

• 0.385 (38.5%)

• 0.090 (9%)

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 PMF는 Probability [ ] Function의 약자이다.

괄호를 채워라

Multiple Select

PMF의 정의로 적합한 것은?

• $$P_X\left(x\right)=\Pr\left(X\le x\right)$$  

•   $$P_X\left(x\right)=\Pr\left(X=x\right)$$  

• $$P_X\left(x\right)=\Pr\left(X>x\right)$$  

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주사위를 12번 던지면 6이 평균 몇 번 나올까?

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Event가 발생할 확률이 p인 Bernoulli 시행을 n번 한다.

단 각 시행은 독립이다.

이 때 발생하는 Event의 수를 Modeling한 확률 변수의 이름은 무엇인가?

[ ] Random Variable (영어 단어를 쓰시오)

Multiple Choice

확률이 p인 Bernoulli 시행을 n번 시행하는 Binomial Random Variable의 평균은?

• $$p$$  

• $$np\left(1-p\right)$$  

• $$np$$   

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6이 나올 때까지 주사위를 계속 던진다고 할 때 평균 몇 번을 던져야 할까?

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Event가 발생할 확률이 p인 Bernoulli 시행을 n번 한다.

단 각 시행은 독립이다.

이 때 발생하는 Event의 수를 Modeling한 확률 변수의 이름은 무엇인가?

[ ] Random Variable (영어 단어를 쓰시오)

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확률이 p인 Bernoulli 시행을 n번 시행하는 Binomial Random Variable의 평균은?

$X\sim B\left(n,p\right),\ E\left[X\right]=?$  

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Event가 발생할 확률이 p인 Bernoulli 시행을

해당 Event가 발생할 때까지 수행한다. 단 각 시행은 서로 독립이다.

Event가 발생할 때 까지 수행한 시행의 수를 Modeling할 때 적합한 확률 변수의 이름은 무엇인가?

[ ] Random Variable (영어 단어를 쓰시오)

Multiple Choice

확률이 p인 Bernoulli 시행을 해당 Event가 발생하지 않을 때까지 반복하여 수행하는 Geometric Random Variable의 평균은 ?

• $$p$$  

• $$\frac{1}{p}$$   

• $$np$$   

Multiple Choice

Alice, Bob, and Carol take turns flipping a fair coin, and the winner is the first player to flip a head. If Alice flips the coin first, what is the probability that Alice wins?

• 1/2

• 7/8

• 4/7

• 3/5

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Binomial 확률 변수 $X\sim B\left(30000,\ \frac{0.61}{30000}\right)$  에 대하여 $P_X\left(1\right)$  의 값을

소수 4번째 자리까지 적으시오.

(Ex: 3.1415)

Multiple Choice

Suppose the arrival of telephone calls at a switch can be modeled with Poisson. That is, if 𝑋 is the number of calls that arrives in 𝑡 minutes, then

$$P_X\left(k\right)=\frac{\left(\lambda\cdot t\right)^k}{k!}e^{-\lambda t}$$  

$$\lambda$$  is the average arrival rate in calls/minute. Suppose that the average rate of calls is 10 per minute

What is the probability that fewer than three calls will be received in the first 6 seconds ?

• $$\approx0.6$$  

• $$\approx0.13527$$  

• $$\approx0.091969$$  

• $$\approx0.00348$$  

Multiple Choice

Suppose the arrival of telephone calls at a switch can be modeled with Poisson. That is, if 𝑋 is the number of calls that arrives in 𝑡 minutes, then

$$P_X\left(k\right)=\frac{\left(\lambda\cdot t\right)^k}{k!}e^{-\lambda t}$$  

$$\lambda$$  is the average arrival rate in calls/minute. Suppose that the average rate of calls is 10 per minute

What is the probability that fewer than three calls will be received in the first 6 minutes ?

• $$\approx9.1969$$  e-2

• $$\approx1.2498$$  e-9

• $$\approx2.1478$$  e-16

• $$\approx1.6295$$  e-