期望值 Expected value

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期望值 Expected value

第 1 節 1st Period

Material Note

Word：Expected value (期望值), Mean (平均值), Weighted Arithmetic Mean (加權平均值), outcome (結果), mystery grab bag(福袋)。

Sentence：

1. We can compute the weighted mean of the mystery grab bags. (我們可以計算福袋的加權平 均數。)

2. We can rewrite this formula into… (將其改寫成…) 3. We can list all of our outcomes and the probability associated with each of those outcomes. (將結果 及其機率列出來。)

Word: discrete (離散的), random (隨機), variable (變 數), converge (收斂), distribution (分配), for short (簡 稱).

Translation:

Set S is a sample space of a trial. A A₁, ₂,,A_n are mutually exclusive events and

 ₁  ₂  _n

S A A A . For any i1,2,,n, each A_i is associated with its probability p_i, and it is

corresponding to m_i, which m_i is a real number.

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Note that

 _{1 1} _{2 2} _{n n} E m p m p m p

is the expected value of this trial.

Another Definition:

Let X be a numerically-valued discrete random variable with sample space  and distribution function m x

 

 

by

   







x

E x xm x , provided this sum converges

absolutely. We often refer to the expected value as the mean, and denote ^{E x}

 

Note: A B is read as A union B.

Translation:

There are three outcomes 10, 20 and 30 dollars by rolling a die one time. The probability associated with each of those outcomes is as follows:

Money 10 20 50

Prob. ³

6

2 6

1 6

By the definition of expected value, we’ve got E of x is equal to 10 dollars times the probability ³ 6 add 20 dollars times the probability ²

6 add 50 dollars times the probability ¹

6 is equal to 20 dollars.

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Word: braised eggs(滷蛋), free-range chicken(土雞) Translation:

We take any 3 braised eggs. There are three outcomes That these braised eggs were produced by free-range chicken 1, 2 and 3. The corresponding probability is as follows:

The number of braised eggs produced by free-range

chicken

0 1 2

Probability 

8 3 10 3

C 56 120 C

 

2 8

1 2

10 3

C C 56 120 C

 

2 8

2 1

10 3

C C 8

120 C

Hence, the expected value for the number of braised eggs produced by free-range chicken is

  56   56   8 3

0 1 2 .

120 120 120 5 E

Note:

1. Cⁿ_k can be written in ⁿC_k, _nC_k, nCk or C n k





2. Cⁿ_k can be read as “n C k” or “n choose k”.

Translation:

There are two possible outcomes within 1 year,

“Life” or “death”, of a 50-year-old man. If the circumstances is “Life”, then the net incomes of insurance company will be 2,400 dollars. Otherwise, the company will pay 2,000,000 – 2,400 (2 million minus 2 thousand and 4 hundred) dollars. The corresponding probability is as follows:

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Customer Life Death Net incomes +2,400 +2,000,000 – 2,400

Probability 0.9998 0.0002

Hence, the expected value for the profit of insurance company is

     

    

 



2400 0.9998 ( 2000000 2400) 0.0002 2400 (0.9998 0.0002) 2000000 0.0002 2400 400

2000 ( ).

E

dollars 補充題

Material

In the casino game roulette, a wheel with 38 spaces (18 red, 18 black, and 2 green) is spun. In one possible bet, the player bets $1 on a single number. If that number is spun on the wheel, then they receive $36 (their original $1 + $35). Otherwise, they

lose their $1. On average, how much money should a player expect to win or lose if they play this game repeatedly?

Solution 1:

Suppose you bet $1 on each of the 38 spaces on the wheel, for a total of $38 bet. When the winning number is spun, you are paid $36 on that number. While you won on that one number, overall you’ve lost $2. On a per-space basis, you have “won”  $2  

$0.053

$38 . In other words, on average you lose 5.3 cents per space you bet on

We call this average gain or loss the expected value of playing roulette. Notice that no one ever loses exactly 5.3 cents: most people (in fact, about 37 out of every 38) lose $1 and a very few people (about 1 person out of every 38) gain $35 (the $36 they win minus the $1 they spent to play the game).

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Solution 2:

There is another way to compute expected value without imagining what would happen if we play every possible space. There are 38 possible outcomes when the wheel spins, so the probability of winning is ¹

38. The complement, the probability of losing, is ³⁷ 38. Summarizing these along with the values, we get this table:

outcome $35 -$1

Probability of outcome ¹ 38

37 38

Notice that if we multiply each outcome by its corresponding probability we get

 1 

$35 0.9211

38 and  37 

$1 0.9737

38 , and if we add these numbers we get

 

   

0.9211 0.9737 0.053, which is the expected value we computed above.

Note

Word: Roulette (輪盤), Spun (spin 旋轉的過去分詞), Summarizing (總結), Complement (補集), Corresponding (對應).

Sentence:

1. On average, how much money should a player expect to win or lose if they play this game repeatedly? (如果他們重覆的玩這個遊戲，一個玩家平均來說會得到多少錢?)

2. Suppose you bet $1 on each of the 38 spaces on the wheel, for a total of $38 bet. (如果你在 輪盤的 38 個格子都下 1 元的注，總共下注 38 元。)

3. The complement, the probability of losing, is ³⁷

38. (它的補集是³⁷

38，也就是輸的機率。)

參考資料 References

1. 許志農、黃森山、陳清風、廖森游、董涵冬（2019）。數學 2：單元 7 期望值。龍騰文 化。

2. DARTMOUTH. (2022, April 26). Grinstead and Snell’s Introduction to Probability.

https://math.dartmouth.edu/~prob/prob/prob.pdf.

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3. OpenTextBookStore. (2022, April 26). Probability.

https://www.opentextbookstore.com/mathinsociety/2.5/Probability.pdf.

製作者：臺北市立陽明高中 吳柏萱 教師