###### tags: `111-2學校上課`
# 機率期中
## Hw2-1
### 題目(英文)
Mr. Smith has **12 shirts**, **eight pairs of slacks**, **eight ties**, and **four jackets**. Suppose that **four shirts, three pairs of slacks, two ties, and two jackets are blue**.
### 題目(中文)
史密斯先生有**12件襯衫**、**8條褲子**、**8條領帶**和**4件夾克**。假設有**4件襯衫**、**3條褲子**、**2條領帶**和**2件夾克**是**藍色**的。
(a) 從隨機選擇中選出全藍色的服裝的概率是多少？
(b) 明天他至少穿一件藍色的物品的概率是多少？
### Ans(a)
先計算出選出來有幾種方法，因為不用管順序故用$\binom{n}{r}$
N(S) = 全部相乘$12*8*8*4=3072$
N(A) = $4*3*2*2=48$
P(A) = $\frac{N(A)}{N(S)}=\frac{48}{3072}=0.015625$
### Ans(b)
先計算出選出來有幾種方法，因為不用管順序故用$\binom{n}{r}$
N(S) = 全部相乘$12*8*8*4=3072$
N(A) = 全部可能性-都沒有藍色的 = 至少一件藍色
N(A$^c$)=$8*5*6*2=480$
N(A) = $3072-480=2592$
P(A) = $\frac{N(A)}{N(S)}=\frac{2592}{3072}=0.84375
## Hw2-2
### 題目(英文)
How many **four-digit numbers** can be formed by using only the digits **2, 4, 6, 8, and 9**? How many of these have **some digit repeated**?
### 題目(中文)
有多少個由數字 2、4、6、8 和 9 組成的**四位數**可以被形成？其中有多少個數字有**某個數字重複**？
### Ans
全部可能性-沒有數字重複=至少一個數字重複

每個地方有五種可能性，四個位子:$5^4$

每個數字都不重複:$5*4*3*2=120每選一個數字後那數字就不能再選了$
$5^4-120=505$
## Hw2-3
### 題目(英文)
Suppose that four cards are drawn successively from an ordinary deck of 52 cards, **with replacement** and **at random**. What is the probability of drawing **at least one king**?
### 題目(中文)
假設從一副有52張牌的普通撲克牌中，**以有放回方式**且**隨機抽取**四張牌。抽到**至少一張國王**的機率是多少？

### Ans
有放回，代表每一次都是沒抽到國王機率=$\frac{48}{52}$
每次獨立 直接相乘=$\frac{48}{52}^4=0.726$
1-{4抽都沒抽到國王}=1-0.726=0274

## Hw2-4
### 題目(英文)
- One of the five elevators in a building leaves the basement with **eight passengers** and stops at all of the remaining **11 floors**.
- If it is equally likely that a passenger gets off at **any of these 11 floors**, what is the **probability that no two of these eight passengers will get off at the same floor**?

### 題目(中文)
某建築物的五台電梯之一帶著八名乘客從地下室開出，接著停靠剩下的11個樓層。如果每位乘客在這11個樓層中下電梯的機率是相等的，則這八位乘客**沒有兩人下電梯在同一層樓**的機率是多少

### Ans
這樣思考，每個人選一個1~11的數字代表樓層，不能重複
先計算所以可能性
$N(S)=11^8人可以選擇11層樓出去$
$N(A)=P^{11}_8$
$P(A)=\frac{P^{11}_8}{11^8} = 0.03103580$

## Hw2-5
### 題目(英文)
The elevator of a **four-floor building** leaves the first floor with **six passengers** and stops at all of the remaining **three floors**. If it is equally likely that a passenger gets off at any of these three floors, what is the probability that, at each stop of the elevator, **at least one passenger departs**?

### 題目(中文)
某棟有四層樓的建築物的電梯在一樓載著六名乘客並停靠剩下的三層樓。如果每位乘客在這三層樓中下電梯的機率是相等的，則在每個停靠點，**至少有一名乘客下電梯**的機率是多少？

### Ans
先計算所以可能性
$N(S)=3^6人可以選擇2~4做為自己要出去的電梯=729$
$N(A)=1-其中一個樓層沒人下$
$N(其中一個樓層沒人下)=3(隨便找一個樓層作為要空的)*每個從剩下樓層找到一個可重複2^6 = 192$
$P(A)=\frac{729-192}{729}=0.736$

## Hw2-5
### 題目(英文)
Let A be the set of all sequences of 0's, 1's, and 2's of length 12. (a) How many elements are there in A? (b) How many elements of A have exactly six 0's and six 1's? (c) How many elements of A have exactly three 0's, four 1's, and five 2's?

### 題目(中文)
讓 A 表示所有長度為 12 的由 0、1 和 2 組成的序列。
(a) A 中有多少個元素？
(b) A 中有多少元素恰好有六個 0 和六個 1？
(c) A 中有多少元素恰好有三個 0，四個 1 和五個 2？

### Ans(a)
$3^{12}$
### Ans(b)
$\frac{12!}{6!*6!}$
### Ans(c)
$\frac{12!}{3!*4!*5!}$

## Hw2-6
### 題目(英文)
Six fair dice are tossed. What is the probability that at least two of them show the same face?

### 題目(中文)
丟六顆公正的骰子。至少有兩顆骰子顯示相同面的機率是多少？

### Ans
$\frac{6}{6^2}=\frac{1}{6}$

## Hw2-7
### 題目(英文)
Six fair dice are tossed. What is the probability that at least two of them show the same face?

### 題目(中文)
丟六顆公正的骰子。至少有兩顆骰子顯示相同面的機率是多少？

### Ans
$\frac{6^6-6!}{6^6}$

## Hw2-8
### 題目(英文)
A fair die is tossed eight times. What is the probability of exactly two 3’s, exactly three 1’s, and exactly two 6’s?

### 題目(中文)
一顆公正的骰子被投擲八次。恰好出現兩次 3、恰好出現三次 1 和恰好出現兩次 6 的機率是多少？

### Ans
$\frac{8!}{2!*3!*2!}=\frac{1}{6}$

## Hw3-1
### 題目(英文)
In a closet there are 10 pairs of shoes. If six shoes are selected at random, what is the probability of (a) no complete pairs; (b) exactly one complete pair?.

### 題目(中文)
在一個櫥櫃裡有 10 對鞋子。如果隨機選擇六只鞋子，
(a) 沒有配對成功的機率是多少？
(b) 恰好有一對配對成功的機率是多少？

### Ans(a)
$\frac{P^{10}_6*2^6}{P^{20}_6}$
### Ans(a)
$\frac{10*P^{9}_4*2!}{P^{20}_6}$
## Hw3-3
### 題目(英文)
A team consisting of three boys and four girls must be formed from a group of nine boys and eight girls. If two of the girls are feuding and refuse to play on the same team, how many possibilities do we have?

### 題目(中文)
從九個男孩和八個女孩中選擇一個由三個男孩和四個女孩組成的隊伍。其中兩個女孩正在鬧矛盾，拒絕在同一個隊伍中玩，問我們有多少種可能性？

### Ans
$\binom{9}{3}*(\binom{6}{4}+\binom{6}{3}*\binom{2}{1})$
## Hw3-4
### 題目(英文)
A fair die is tossed six times. What is the probability of getting exactly two 6’s?

### 題目(中文)
一個公正的骰子被投擲六次。得到正好兩個6的機率是多少？

### Ans
$\frac{\binom{6}{2}*5^4}{6^6}$

## Hw3-5
### 題目(英文)
By a combinatorial argument, prove that for r ≤ n and r ≤ m,
$\binom{n+m}{r}=\binom{n}{0}*\binom{m}{r}....\binom{n}{n-r}*\binom{m}{0}$

### 題目(中文)
證明

### Ans
這個等式表明，在從 n 個紅色球和 m 個藍色球中選擇 r 個球時，我們必須選擇 r 個紅球和 0 個藍球，或者選擇 r - 1 個紅球和 1 個藍球，或者選擇 r - 2 個紅球和 2 個藍球，以此類推，直到選擇 0 個紅球和 r 個藍球為止。

## Hw3-6
### 題目(英文)
There are 12 balls in a box. Among these balls, two balls are yellow. Someone withdrew 7 balls randomly from the box. What is the probability that the remaining 5 balls contains at least 1 yellow ball?

### 題目(中文)
一個盒子中有12個球，其中有兩個是黃色的。有人從盒子中隨機取出七個球。那麼剩下的五個球中至少有一個黃球的機率是多少？

### Ans
$\frac{\binom{2}{1}*\binom{10}{6}+\binom{10}{7}}{\binom{12}{7}}$

## Hw3-7
### 題目(英文)
If eight defective and 12 nondefective items are inspected one by one, at random and without replacement, what is the probability that
(a) the first four items inspected are defective;
(b) from the first three items at least two are defective?

### 題目(中文)
如果隨機且不重複地檢查八個有缺陷和十二個無缺陷的物品，那麼：

(a) 前四個檢查的物品都是有缺陷的機率是多少？

(b) 從前三個檢查的物品中，至少兩個是有缺陷的機率是多少？

### Ans(a)
$\frac{\binom{8}{4}}{\binom{20}{4}}$
### Ans(a)
$\frac{\binom{8}{2}*\binom{12}{2}+\binom{8}{3}*\binom{12}{1}+\binom{8}{4}*\binom{12}{0}}{\binom{20}{3

## Hw3-8
### 題目(英文)
From an ordinary deck of 52 cards, cards are drawn one by one, at random and without replacement. What is the probability that the fourth heart is drawn on the tenth draw? Hint: Let F denote the event that in the first nine draws there are exactly three hearts, and E be the event that the tenth draw is a heart. Use P(FE) =P(F)P(E | F)

### 題目(中文)
從一副普通的 52 張牌的牌組中，按照隨機且不重複的方式一張一張地抽出牌。問第十次抽牌時第四個紅心被抽出的機率是多少？提示：令 F 表示前九次抽牌中恰好有三張紅心的事件，E 表示第十次抽牌抽到紅心的事件。使用 $P(FE) = P(F)P(E | F)。$

### Ans
$P(FE) = P(F)P(E | F)=>\frac{P(FE)}{P(E | F)}= P(F)$
$P(F) = (C(13,3) * C(39,6)) / C(52,9)$
$P(E|F) = 從43張找到最後一張紅心= \frac{10}{43}$

## Hw3-9
### 題目(英文)
Suppose that 40% of the students of a campus are men. If 20% of the men and 16% of the women of this campus own driver licenses, what percent of all of them own driver licenses?

### 題目(中文)
假設一個校園裡有40%的學生是男性。如果男性中有20％和女性中有16％擁有駕照，那麼所有學生中擁有駕照的百分比是多少？

### Ans
$0.4*0.2+0.6*0.84$
## Hw3-10
### 題目(英文)
One of the cards of an ordinary deck of 52 cards is lost. What is the probability that a random card drawn from this deck is a spade?

### 題目(中文)
從一副普通的52張牌中有一張牌遺失了，從這副牌中隨機抽出一張牌，該牌是黑桃牌的概率是多少？

### Ans
$\frac{13}{52}\cdot\frac{3}{4} + \frac{12}{52}\cdot\frac{1}{4}$

## Hw3-11
### 題目(英文)
A box contains 18 tennis balls, of which eight are new. Suppose that three balls are selected randomly, played with, and after play are returned to the box. If another three balls are selected for play a second time, what is the probability that they are all new

### 題目(中文)
一個盒子裡有18個網球，其中8個是新的。假設隨機選擇3個球進行比賽，比賽結束後放回盒中。如果第二次再選擇3個球進行比賽，那麼它們全是新球的概率是多少？

### Ans
$\frac{\binom{8}{3}*\binom{5}{3}+\binom{8}{2}*\binom{10}{1}*\binom{6}{3}+\binom{8}{1}*\binom{10}{2}}{\binom{18}{3}*\binom{18}{3}}$

## Hw4-7
### 題目(英文)
An experiment consists of first tossing a fair coin and then drawing a card randomly from an ordinary deck of 52 cards with replacement. If we perform this experiment successively, what is the probability of obtaining heads on the coin before an ace from the cards?

### 題目(中文)
這個實驗包括先投擲一枚公正硬幣，然後從一副有52張牌的普通撲克牌中有放回地隨機抽一張牌。如果我們連續進行這個實驗，那麼在從牌中抽到一張A之前先得到正面的機率是多少？
### Ans