diff --git a/00035_Expected_Values/en_article_proofreading.md b/00035_Expected_Values/en_article_proofreading.md
index a1dd8de..3268041 100644
--- a/00035_Expected_Values/en_article_proofreading.md
+++ b/00035_Expected_Values/en_article_proofreading.md
@@ -3,152 +3,153 @@ keywords:
- probability
- expected value
- geometric sequence
-is_finished: False
+is_finished: True
remark: Note for translators. Instead of the amount in CZK, you can enter the currency of your country, or some reasonable currency. You can also come up with your own names for the lottery tickets.
---
-# Which lottery ticket is more profitable?
+# Which Lottery Ticket Is More Profitable?
-Very often in life we find ourselves in situations where chance and probability are involved.
-Imagine being faced with a choice between several options, e.g., when choosing a lottery ticket or investing in a project.
-Each choice has its risks and potential rewards, but the question is, how to find out which one is the most advantageous?
+In life we very often encounter situations that involve chance and probability.
+Imagine being faced with a choice between several options—for example, when choosing a lottery ticket or investing in a project.
+Each choice has its risks and potential rewards, but how can we determine which one is the most advantageous?
This is where the so-called *expected value* comes into play.
-Expected value tells us what outcome we can expect on average if we choose a particular option.
-It helps us better estimate what will pay off in the long run.
+Expected value tells us the average outcome we can anticipate when choosing a particular option.
+It helps us better estimate which option is likely to pay off in the long run.
It is not an exact prediction, but a tool that allows us to better understand risk and reward, both in simple games and in real life decisions.
-For example, consider two lottery numbers:
+Let's consider two lottery tickets, for example:
-* Lot A: It costs 10 CZK and with probability $0{,}1$ we win 100 CZK, otherwise we win nothing.
-* Lot B: It costs 10 CZK and with probability $0{,}2$ we win 60 CZK, otherwise we win nothing.
+* Ticket A: It costs 10 CZK and has a $0{.}1$ probability of winning 100 CZK; otherwise, it wins nothing.
+* Ticket B: It costs 10 CZK and has a $0{.}2$ probability of winning 60 CZK; otherwise, it wins nothing.
-For ticket A, we expect that if we buy 10 tickets, one of them will win 100 CZK and nine will win nothing.
-So we can expect that each lottery ticket will bring us 10 CZK on average.
+For ticket A, we expect that if we buy 10 tickets, one of them will win 100 CZK while the reaming nine will win nothing.
+Therefore, we can expect that each lottery ticket will yield an average return of 10 CZK.
Similarly, for lottery ticket B, we expect that if we buy 10 tickets, two of them will win 60 CZK and eight will win nothing.
-We can therefore expect each lottery ticket to bring us 12 CZK on average.
+We can therefore expect each lottery ticket to yield an average return of 12 CZK.
-Therefore the lottery ticket B is more profitable.
+This shows that ticket B is the better option.
-## Expected value
+## Expected Value
-The average win we just calculated is called the *expected value* (or also the *mean value*).
+The average win we just calculated is called the *expected value*.
In general, we can say that for a random variable $X$ that takes on finitely many values $x_1,\,\dots,\,x_k$ with probabilities $p_1,\,\dots,\,p_k$,
-we calculate its expected value
+we calculate its expected value using this formula:
$$
EV=\sum_{i=1}^k x_ip_i.
$$
-## Which lottery ticket is the best?
+## Which Lottery Ticket Is the Best?
Let's take a look at three lottery tickets.
-A black pearl worth 50 CZK, the Black Pearl worth 100 CZK and the Rentiér lottery ticket worth 50 CZK.
+The 50 CZK Black Pearl ticket, the 100 CZK Black Pearl ticket and the Rental King lottery ticket worth 50 CZK.
-The prize structure for the 50 CZK Black Pearl lottery tickets, of which there are $13,000,000 in total, is as follows as follows.
+The prize structure for the 50 CZK Black Pearl lottery tickets, of which there are 13,000,000 in total, is as follows.
-| Amount of money won on the lottery ticket (in CZK) | Number of winning tickets |
+| Prize amount (in CZK) | Number of winning tickets |
| ------------- | ------------- |
-| $50$ | $1\,820\,000$ |
-| $100$ | $1\,040\,000$ |
-| $150$ | $260\,000$ |
-| $200$ | $130\,000$ |
-| $300$ | $130\,000$ |
-| $500$ | $104\,000$ |
-| $1\,000$ | $5\,550$ |
-| $2\,000$ | $2\,300$ |
-| $4\,000$ | $480$ |
-| $10\,000$ | $185$ |
-| $20\,000$ | $84$ |
-| $100\,000$ | $14$ |
-| $1\,500\,000$ | $6$ |
-| Celkem | $3\,492\,619$ |
-
-The prize structure for the 100 CZK Black Pearl lottery ticket looks similar, with a total of $15,000,000 issued.
-
-| Amount of money won on the lottery ticket (in CZK) | Number of winning tickets |
+| $50$ | $1{,}820{,}000$ |
+| $100$ | $1\,040{,}000$ |
+| $150$ | $260{,}000$ |
+| $200$ | $130{,}000$ |
+| $300$ | $130{,}000$ |
+| $500$ | $104{,}000$ |
+| $1{,}000$ | $5{,}550$ |
+| $2{,}000$ | $2{,}300$ |
+| $4{,}000$ | $480$ |
+| $10{,}000$ | $185$ |
+| $20{,}000$ | $84$ |
+| $100{,}000$ | $14$ |
+| $1{,}500{,}000$ | $6$ |
+| Total | $3{,}492{,}619$ |
+
+The prize structure for the 100 CZK Black Pearl lottery ticket looks similar, with a total of $15{,}000{,}000$ issued tickets.
+
+| Prize amount (in CZK) | Number of winning tickets |
| ------------- | ------------- |
-| $100$ | $2\,400\,000$ |
-| $200$ | $900\,000$ |
-| $300$ | $450\,000$ |
-| $500$ | $150\,000$ |
-| $600$ | $150\,000$ |
-| $900$ | $75\,000$ |
-| $1\,000$ | $75\,000$ |
-| $1\,500$ | $20\,000$ |
-| $6\,000$ | $4\,000$ |
-| $20\,000$ | $185$ |
-| $50\,000$ | $84$ |
-| $100\,000$ | $30$ |
-| $200\,000$ | $13$ |
-| $5\,000\,000$ | $6$ |
-| Celkem | $4\,224\,318$ |
-
-For the 3rd and final round, let's take a look at the Rentiér lottery ticket,
-of which $8,000,000 has been issued and the prizes are shown in the table below.
-
-| Amount of money won on the lottery ticket (in CZK) | Number of winning tickets |
+| $100$ | $2{,}400{,}000$ |
+| $200$ | $900{,}000$ |
+| $300$ | $450{,}000$ |
+| $500$ | $150{,}000$ |
+| $600$ | $150{,}000$ |
+| $900$ | $75{,}000$ |
+| $1{,}000$ | $75{,}000$ |
+| $1{,}500$ | $20{,}000$ |
+| $6{,}000$ | $4{,}000$ |
+| $20{,}000$ | $185$ |
+| $50{,}000$ | $84$ |
+| $100{,}000$ | $30$ |
+| $200{,}000$ | $13$ |
+| $5{,}000{,}000$ | $6$ |
+| Total | $4{,}224{,}318$ |
+
+Last but not least, let's take a look at the Rental King lottery ticket,
+with a total of $8{,}000{,}000$ tickets issued. The prizes are shown in the table below.
+
+| Prize amount (in CZK) | Number of winning tickets |
| ------------- | ------------- |
-| $50$ | $960\,000$ |
-| $100$ | $720\,000$ |
-| $150$ | $160\,000$ |
-| $250$ | $160\,000$ |
-| $500$ | $70\,000$ |
-| $1\,000$ | $1\,300$ |
-| $2\,000$ | $500$ |
-| $5\,000$ | $160$ |
-| $10\,000$ | $80$ |
-| $100\,000$ | $6$ |
-| $3\,500\,000$ | $3$ |
-| Celkem | $2\,072\,049$ |
-
-The top prize of $3\,500\,000\,\text{Kč}$ is not paid at once, but consists of an immediate prize of $500\,000\,\text{Kč}$
-and an annuity of $50\,000\,\text{Kč}$ for 5 years.
-
-> **Exercise 1.** Which of the above ticket has the highest chance of winning?
+| $50$ | $960{,}000$ |
+| $100$ | $720{,}000$ |
+| $150$ | $160{,}000$ |
+| $250$ | $160{,}000$ |
+| $500$ | $70{,}000$ |
+| $1{,}000$ | $1{,}300$ |
+| $2{,}000$ | $500$ |
+| $5{,}000$ | $160$ |
+| $10{,}000$ | $80$ |
+| $100{,}000$ | $6$ |
+| $3{,}500{,}000$ | $3$ |
+| Total | $2{,}072{,}049$ |
+
+The top prize of $3{,}500{,}000\,\text{CZK}$ is not paid at once, but consists of an immediate prize of $500{,}000\,\text{CZK}$
+and an annuity of $50{,}000\,\text{CZK}$ for 5 years.
+
+> **Exercise 1.** Which ticket has the highest chance of winning?
\iffalse
-*Solution.* In the case of the Black Pearl ticket for 50 CZK, out of the total number of 13\,000\,000$ pieces,
-there are $3\,492\,619$ winning tickets (see the last row of the table).
+*Solution.* In the case of the 50 CZK Black Pearl ticket, there are $3{,}492{,}619$ winning tickets
+out of the total number of $13{,}000{,}000$ (see the last row of the table).
The probability that a randomly selected ticket is a winning one can be calculated as
$$
-P(V_1)=\frac{3\,492\,619}{13\,000\,000}=0{,}268633\,.
+P(V_1)=\frac{3{,}492{,}619}{13{,}000{,}000}=0{.}268633.
$$
-We can say that if we buy one lottery ticket, we have a chance of winning about $26{,}86\,\%$.
-By adjusting the fraction, we can also find that the chance of getting a winning lottery ticket is $1\colon 3{,}72$.
-Similarly, in the case of the Black Pearl lottery ticket, worth $100, we get
+We can say that if we buy one lottery ticket, we have a chance of winning about $26{.}86\%$.
+By manipulating the fraction, we can also see that the chance of getting a winning lottery ticket is $1\colon 3{.}72$.
+
+Similarly, in the case of the 100 CZK Black Pearl lottery ticket we get
$$
-P(V_2)=\frac{4\,224\,318}{15\,000\,000}=0{,}2816212\,.
+P(V_2)=\frac{4{,}224{,}318}{15{,}000{,}000}=0{.}2816212.
$$
-That is, the chance of winning is $28{,}16\,\%$ or $1\colon 3{,}55$.
+That means that the chance of winning is $28{.}16\%$ or $1\colon 3{.}55$.
-In the case of the Rentier ticket, we have
+In the case of the Rental King ticket, we get
$$
-P(V_3)=\frac{2\,072\,049}{8\,000\,000}=0{,}259\,,
+P(V_3)=\frac{2{,}072{,}049}{8{,}000{,}000}=0{.}259,
$$
-so the chance of winning is $25{,}9{,}%$ or $1\colon 3{,}86$.
+so the chance of winning is $25{.}9\%$ or $1\colon 3{.}86$.
-Comparing the individual probabilities of winning, we see that the greatest chance of winning is when buying a Black Pearl ticket worth 100 CZK.
+Comparing the individual probabilities of winning, we find that the highest chance of winning comes with the 100 CZK Black Pearl ticket.
-In this context, we can also consider what we call a winning ticket.
-A winning ticket is one that has a prize.
-But if we paid 100 CZK for the ticket, then a win of 100 CZK will pay us back, but we haven't actually won anything.
-In order to get the probability of actually winning, we won't consider the first row in our winning tables.
+In this context, we can also consider what we mean by a winning ticket.
+A ticket is usually considered winning if it yields any amount of money.
+But when we paid 100 CZK for the ticket, then a win of 100 CZK will pay us back, but we haven't actually won anything.
+In order to get the probability of actually winning more than we paid, we won't consider the first row in our winning tables.
This way we get adjusted winning probabilities
$$
\begin{aligned}
-P(V_1)&=\frac{1\,672\,619}{13\,000\,000}=0{,}128633\\
-P(V_2)&=\frac{1\,824\,318}{15\,000\,000}=0{,}1216212\\
-P(V_3)&=\frac{1\,112\,049}{8\,000\,000}=0{,}139\,.
+P(V_1)&=\frac{1{,}672{,}619}{13{,}000{,}000}=0{.}128633,\\
+P(V_2)&=\frac{1{,}824{,}318}{15{,}000{,}000}=0{.}1216212,\\
+P(V_3)&=\frac{1{,}112{,}049}{8{,}000{,}000}=0{.}139.
\end{aligned}
$$
-We can see that if we consider the lottery tickets that actually win more than their cost,
-the best lottery ticket is the Rentiér lottery ticket, where the chance of winning is $13{,}9\,\%$.
+We can see that when we only consider the lottery tickets that return more than their purchase cost,
+the best option is the Rental King lottery ticket that has a $13{.}9\%$ chance of such win.
\fi
@@ -156,100 +157,99 @@ the best lottery ticket is the Rentiér lottery ticket, where the chance of winn
\iffalse
-*Řešení.* To calculate the expected value, by definition, we need to know the probabilities of individual wins:
+*Solution.* To calculate the expected value, by definition, we need to know the probabilities of individual wins:
-| Amount won on the lottery ticket (in CZK) | Probability of a given winning |
+| Prize amount (in CZK) | Winning probability |
| ------------- | ------------- |
-| $50$ | $0{,}14$ |
-| $100$ | $0{,}08$ |
-| $150$ | $0{,}02$ |
-| $200$ | $0{,}01$ |
-| $300$ | $0{,}01$ |
-| $500$ | $0{,}008$ |
-| $1\,000$ | $0{,}000\,426\,9$ |
-| $2\,000$ | $0{,}000\,176\,923$ |
-| $4\,000$ | $0{,}000\,036\,923$ |
-| $10\,000$ | $0{,}000\,014\,231$ |
-| $20\,000$ | $0{,}000\,006\,461\,538$ |
-| $100\,000$ | $0{,}000\,006\,461\,538$ |
-| $1\,500\,000$ | $0{,}000\,000\,461\,538$ |
-
-If we denote the values of individual winnings $n_1$ to $n_{13}$ and their corresponding probabilities $p_1$ to $p_{13}$,
-we get the expected value $EV(L_1)$ of the Black Pearl ticket
+| $50$ | $0{.}14$ |
+| $100$ | $0{.}08$ |
+| $150$ | $0{.}02$ |
+| $200$ | $0{.}01$ |
+| $300$ | $0{.}01$ |
+| $500$ | $0{.}008$ |
+| $1{,}000$ | $0{.}0004269$ |
+| $2{,}000$ | $0{.}000176923$ |
+| $4{,}000$ | $0{.}000036923$ |
+| $10{,}000$ | $0{.}000014231$ |
+| $20{,}000$ | $0{.}000006461538$ |
+| $100{,}000$ | $0{.}000006461538$ |
+| $1{,}500{,}000$ | $0{.}000000461538$ |
+
+Let the values of individual winnings be denoted by $n_1$ to $n_{13}$ and their corresponding probabilities $p_1$ to $p_{13}$.
+Then, the expected value $EV(L_1)$ of the Black Pearl ticket is
$$
-EV(L_1)=\sum_{k=1}^{13}n_kp_k=29\,\text{Kč}.
+EV(L_1)=\sum_{k=1}^{13}n_kp_k=29\,\text{CZK}.
$$
-Given how the individual probabilities are calculated, we can also calculate the expected value as follows
+Given how the individual probabilities are calculated, we can also calculate the expected value as follows:
$$
-EV(L_1)=\frac{1}{13\,000\,000}\left(50\cdot 1\,820\,000+100\cdot1\,040\,000+ \cdots + 100\,000\cdot14+1\,500\,000\cdot6 \right).
+EV(L_1)=\frac{1}{13{,}000{,}000}\left(50\cdot 1{,}820{,}000+100\cdot1{,}040{,}000+ \cdots + 100{,}000\cdot14+1{,}500{,}000\cdot6 \right).
$$
-This approach is preferable as we do not have to calculate the probability of each possible win in the win table.
-For the 100 CZK worth Black Pearl ticket, we get the expected value $EV(L_2)$:
+This approach is more convinient because we don't have to calculate the probability of each possible prize in the table.
+For the 100 CZK Black Pearl ticket, we get the expected value $EV(L_2)$:
$$
-EV(L_2)=\frac{1}{15\,000\,000}\left(100\cdot 2\,400\,000+200\cdot 900\,000+ \cdots + 200\,000\cdot 13+5\,000\,000 \cdot 6 \right)=64\,\text{Kč}.
+EV(L_2)=\frac{1}{15{,}000{,}000}\left(100\cdot 2{,}400{,}000+200\cdot 900{,}000+ \cdots + 200{,}000\cdot 13+5{,}000{,}000 \cdot 6 \right)=64\,\text{CZK}.
$$
-And for the Rentiér lottery ticket we get the expected value $EV(L_3)$:
+And for the Rental King lottery ticket we get the expected value $EV(L_3)$:
$$
-EV(L_3)=\frac{1}{8\,000\,000}\left(50\cdot 960\,000+100\cdot 720\,000+ \cdots + 100\,000\cdot 6+3\,500\,000\cdot 3 \right)=29{,}25\,\text{Kč}.
+EV(L_3)=\frac{1}{8{,}000{,}000}\left(50\cdot 960{,}000+100\cdot 720{,}000+ \cdots + 100{,}000\cdot 6+3{,}500{,}000\cdot 3 \right)=29{.}25\,\text{CZK}.
$$
*Note.*
-* Usually, lotteries state the total amount of winnings and the number of tickets, the expected value is of course the ratio of these two numbers.
-* The stated values are often even lower in reality, as tax is often paid on winnings.
+* Lotteries usually state the total prize pool and the number of tickets. The expected value is, of course, the ratio of these two numbers.
+* The stated values are often even lower in reality, since lottery prizes are usually taxed.
* The same approach can be used to calculate the expected value of a pack of various trading card games (Pokémon, Lorcana, Magic the Gathering or sports cards).
\fi
-> **Exercise 3.** In the previous examples, we considered the main prize of the Rentiér lottery worth $3\,500\,000\,\text{CZK}$.
-> But is this really the real value of the prize, given that it is not paid out all at once?
+> **Exercise 3.** In the previous examples, we considered the top prize of the Rental King lottery to be $3{,}500{,}000\,\text{CZK}$.
+> But is this really the actual value of the prize, given that it is not paid out all at once?
\iffalse
*Solution.* The simple answer is that it is not.
It is important to remember that if we got the money immediately, we could save it or invest it somehow.
-To find out what the value of 50 000 CZK that we will receive in a month is, we can use a concept called *present value*.
-When using it, we ask ourselves how much money we would have to invest today to get the amount we want in a month (e.g. considered 50 000 CZK).
-And this value is then the so-called present value.
+To determine the value of $50{,}000\,\text{CZK}$ recieved one month from now, we can use a concept known as *present value*.
+When using this concept, we ask ourselves how much money we would have to invest today to get the amount we want in one month (e.g. considered $50{,}000\,\text{CZK}$).
+This amount is what we call the present value.
-Suppose we could save the given amount $P_0$ for a month with a monthly interest rate of $0{,}5\,\%$.
-We would then get $P_1=1{,}005P_0$ per month. The present value is then the amount $P_0$, which we must deposit so that $P_1$ is $50\,000\,\text{CZK}$, i.e.
+Let's suppose we could save the given amount $P_0$ for a month with a monthly interest rate of $0{.}5\%$.
+We would then get $P_1=1{,}005P_0$ after one month. The present value is then the amount $P_0$ that we must deposit so that $P_1$ is $50{,}000\,\text{CZK}$, i.e.
$$
-P_0=\frac{50\,000}{1{,}005}=49\,751{,}24\,\text{Kč}.
+P_0=\frac{50{,}000}{1{.}005}=49{,}751{.}24\,\text{CZK}.
$$
-If we want to determine the present value of an amount that we will receive in $n$ months,
+To determine the present value of an amount to be received in $n$ months,
we assume that we will keep the given amount deposited for the entire time.
-We then use compound interest and get the present value $P_0$ of the amount $P_n$ that we will receive in $n$ months as
+Using compound interest, we obtain the present value $P_0$ of the amount $P_n$ recieved after $n$ months as follows:
$$
-P_0=\frac{P_n}{1{,}005^n}.
+P_0=\frac{P_n}{1{.}005^n}.
$$
-Recall that the main prize of the Rentiér lottery consists of $500\,000\,\text{CZK}$ and thirty monthly payments of $50\,000\,\text{CZK}$.
-Considering a monthly interest rate of $0{,}5\,\%$, the present value $PV$ of these payments is
+Let us recall that the top prize of the Rental King lottery consists of $500{,}000\,\text{CZK}$ and thirty monthly payments of $50{,}000\,\text{CZK}$.
+Considering a monthly interest rate of $0{.}5\,\%$, the present value $PV$ of these payments is
$$
-PV=\frac{50\,000}{1{,}005}+\frac{50\,000}{1{,}005^2}+\cdots+\frac{50\,000}{1{,}005^{29}}+\frac{50\,000}{1{,}005^{30}}\,.
+PV=\frac{50{,}000}{1{.}005}+\frac{50{,}000}{1{.}005^2}+\cdots+\frac{50{,}000}{1{.}005^{29}}+\frac{50{,}000}{1{.}005^{30}}.
$$
-We can notice that this is the sum of the terms of a geometric sequence and thus the calculation can be significantly shortened.
+Notice that this is the sum of the terms of a geometric sequence and thus the calculation can be significantly shortened.
$$
-PV=\frac{50\,000}{1{,}005}\cdot\frac{1-\left(\frac{1}{1{,}005}\right)^{30}}{1-\frac{1}{1{,}005}}=1\,389\,702{,}7\,\text{Kč}.
+PV=\frac{50{,}000}{1{.}005}\cdot\frac{1-\left(\frac{1}{1{.}005}\right)^{30}}{1-\frac{1}{1{.}005}}=1{,}389{,}702{.}7\,\text{CZK}.
$$
-Therefore we assume that the value of the main prize is only $1\,889\,702{,}7\,\text{Kč}$.
+Therefore, we assume that the value of the top prize is only $1{,}889{,}702{.}7\,\text{CZK}$.
-If we use this amount to calculate to expected value of the Rentier lottery, we get
-(nebo: If we adjust the calculation of the expected value of the Rentier lottery ticket using this amount, we get.)
+When we use this amount to calculate the expected value of the Rental King lottery ticket, we get
$$
-EV(L_3)=28{,}65\,\text{Kč}.
+EV(L_3)=28{.}65\,\text{CZK}.
$$
*Note.* Previous considerations were still very simplistic, as they did not include, for example, the effect of inflation.
\fi
-> **Exercise 4.** Based on the results of the previous tasks, choose the lottery ticket that is most advantageous.
+> **Exercise 4.** Based on the results of the previous tasks, choose the best lottery ticket.
\iffalse
@@ -257,22 +257,22 @@ $$
1. By probability of winning.
-According to this criterion, the best ticket is the Černá perla ticket worth 100 CZK, which has a chance of winning $28{,}16\,\%$,
-then the Black Pearl ticket worth 50 CZK with a chance of $26{,}86\,\%$ and the worst is the Rentiér ticket with a chance of $25{,}9\,\%$.
+According to this criterion, the best ticket is the 100 CZK Black Pearl with a $28{.}16\%$ chance of winning,
+followed by the 50 CZK Black Pearl ticket with $26{.}86\%$ a chance of winning and the worst is the Rental King ticket with a chance of winning at $25{.}9\%$.
2. By probability of actual winning.
-If we consider the chance of winning more than we paid, we get the following other order.
-The best is the Rentiér ticket with a chance of winning $13{,}9\,\%$, then the Černá perla ticket worth 50 CZK with a chance of $12{,}86\,\%$
-and the last is the Black Pearl ticket worth 100 CZK with a chance of winning $12{,}16\,\%$.
+When we instead consider the chance of winning more than we paid, we get a different ranking.
+The best is the Rental King ticket with a $13{.}9\%$ chance of winning, then the 50 CZK Black Pearl ticket with a $12{.}86\%$ chance of winning
+and the last is the 100 CZK Black Pearl ticket with a $12{.}16\%$ chance of winning.
3. By expected value.
-The expected value of a 50 CZK Black Pearl ticket is 29 CZK. On average we lose 21 CZK on one ticket.
-Similarly, the expected value of a 100 CZK Black Pearl ticket is 64 CZK. On average we lose 36 CZK.
-And in the case of a 50 CZK Rentiér ticket, the adjusted expected value is 28,65 CZK, so on average we lose 21,35 CZK.
+The expected value of the 50 CZK Black Pearl ticket is $29\,\text{CZK}$. On average, we lose $21\,\text{CZK} buying one ticket.
+Similarly, the expected value of the 100 CZK Black Pearl ticket is $64\,\text{CZK}. On average we lose $36\,\text{CZK}.
+And in the case of the 50 CZK Rental King ticket, the adjusted expected value is $28{.}65\,\text{CZK}, so on average we lose $21{.}35\,\text{CZK}.
-We can say that (expectedly) all tickets are losing. But we can consider the 50 CZK Black Pearl ticket to be the best, as it is the least loss-making.
+We can therefore say that, as expected, all of the tickets result in a loss. However, the 50 CZK Black Pearl ticket can be considered the best, since it yields the smallest loss.
\fi