Exploring Expected Values
Introduction
Expected values can be a little tricky to understand the first time around. Using some simple and more advanced examples, this article will help you understand the mystery behind Expected Values. We will specifically answer the following questions with the help of R:- Q1: How much do I gain - or lose - on average if I repeatedly play a given gambling game?
- Q2: How much can I expect to gain - or lose - by making a particular bet?
- Q3: What is the likelihood of owing your friend money after 3 bets given 4:1 favorable odds?
- Q4: What is the minimum number of bets needed to have a ~98% chance of a net win given 4:1 odds?
Gratitude & Motivation
Inspiration and the early examples used for this post come directly from Josh Starmer’s StatQuest video on Expected Values. I really enjoy Josh’s delivery and highly recommend his videos.Expected Value Defined
Expected Values and Probability are two essential foundations of statistics. According to the StatLect website, Expected Values were “first devised in the 17th century to analyze gambling games and answer questions such as:- How much do I gain - or lose - on average, if I repeatedly play a given gambling game
- How much can I expect to gain - or lose - by making a certain bet?
If the possible outcomes of the game (or the bet) and their associated probabilities are described by a random variable, these questions can be answered by computing its expected value.
The expected value is a weighted average of the possible realizations of the random variable (the possible outcomes of the game). Each realization is weighted by its probability” - Marco Taboga, Ph.D.
Load Libraries
Before we get started, let’s load some libraries.library(tidyverse)
Setting Up the Data Set and Probabilities
Let’s assume that we randomly asked 200 individuals if they preferred vanilla or strawberry ice cream. After the survey, there were 40 vanilla fans and 160 strawberry fans.
After tallying the results, we can calculate the probability of a person preferring either vanilla or strawberry ice cream, as shown below.
data_set <- data_set %>% mutate(total_n_size = sum(n_size), probability = round(n_size/total_n_size,4)) data_set
The calculation above shows that 20% of the people surveyed prefer vanilla, and 80% prefer strawberry.
For this example, let’s assume the survey results represent the population. Therefore, we can expect very similar outcomes if we surveyed another random group of 200 individuals.
Betting with Emotions Over Facts
Now let’s assume that your friend is a giant vanilla ice cream fan and does not want to believe the survey results. He is convinced that even though the stats say something different, he feels more people prefer vanilla over strawberry.In fact, your friend is so confident that he wants to wager a bet on it. He tells you that for the following 100 people surveyed, he will give you $1 for every strawberry fan, and you will have to pay him $1 for every vanilla fan.
Remember, 80% of people prefer strawberry, and only 20% prefer vanilla. The odds are heavily in your favor, and this sounds like a good bet. However, you hate to lose money, even if it’s only $1. Therefore, let’s first answer the following questions before you make a bet.
Q1: How much do I gain - or lose - on average if I repeatedly play a given gambling game?
Remember that about 80% of people are strawberry fans, and about 20% are vanilla fans. These odds are in your favor.
Since your friend is a huge fan of vanilla ice cream and is betting with emotion versus facts, we assume this is a good bet. In fact, on average, we calculate that you would make about $0.60 per bet or approximately $60 if we bet 100 times in a row. See a breakdown of the math below.
- Estimated Dollars Lost = $1 x 20% x 100 bets = $20 lost
- Estimated Dollars Won = $1 x 80% x 100 bets = $80 won
- Final Tally = $80 - $20 = $60
- Estimated amount earned per bet = $60 / 100 bets = $0.60 per bet
outcome <- tibble(outcome = c(-1, 1)) data_set <- cbind(data_set, outcome) %>% mutate(expected_prob = probability * outcome, num_new_bets = 100, expected_value = expected_prob * num_new_bets) glimpse(data_set)
Q2: How much can I expect to gain - or lose - by making a certain bet?
Technically we answered this question if 100 bets were made. We can expect to win about $0.60 every time we bet. Remember, this assumes multiple bets, and although we are betting $1 for each bet, the $0.60 represents winnings after averaging numerous iterations.
certain_bet_amount <- data_set %>% summarise(certain_bet = sum(expected_value) / max(num_new_bets)) certain_bet_amount
Q3: What is the likelihood of owing your friend money after 3 bets given 4:1 favorable odds?
However, instead of asking 100 random people their opinion, what if your friend only wants to ask the next 3 random people if they prefer vanilla or strawberry? Given the probabilities above, the odds of you owing your friend money is ~10.2% if you bet only 3 times. Let’s prove it out with 1,000 experiments.
We will uniformly assign 3 random values between 0 and 1 for each of the 1,000 experiments to prove it out. If one of the random numbers is less than 0.20, you lose $1; otherwise, you gain $1. Remember, 0.20 represents the probability that someone prefers vanilla ice cream.
We will tally up the totals for each experiment and count how many times you were “in the red” versus “in the black”.
set.seed(1324) three_bet_experiment <- tibble( experiment_num = rep(c(1:1000), each = 3), random_value = runif(3000, 0, 1)) three_bet_experiment <- three_bet_experiment %>% mutate(win_loss = case_when(random_value <= 0.20 ~ -1, TRUE ~ 1))
Let’s take a look at how the data set looks before we summarize the results. Keep in mind that each experiment has a total of 4 possible outcomes since the order does not matter:
- WWW = Win a total of $3
- WWL | WLW | LWW = Win a total of $1
- LLW | LWL | WLL = Lose a total of $1
- LLL = Lose a total of $3
head(three_bet_experiment)
- Experiment #1 Results: LWW = $1 net win
- Experiment #2 Results: WLW = $1 net win
three_bet_aggr <- three_bet_experiment %>% group_by(experiment_num) %>% summarise(win_loss_sum = sum(win_loss)) %>% ungroup() %>% mutate(win_loss_bucket = case_when(win_loss_sum < 0 ~ str_c("Loss of $", abs(win_loss_sum)), TRUE ~ str_c("Winnings of $", win_loss_sum))) %>% mutate(win_loss_tally = 1) head(three_bet_aggr)
To set up the bar chart, let’s aggregate the results.
three_bet_graph <- three_bet_aggr %>% select(win_loss_bucket, win_loss_tally) %>% group_by(win_loss_bucket) %>% summarise(win_loss_tally = sum(win_loss_tally)) %>% ungroup() %>% mutate(win_loss_perc = win_loss_tally / 1000) head(three_bet_graph)
Next, we can visualize how each of those betting scenarios played out over 1,000 experiments. Remember, there are only 4 possible net outcomes for each experiment. You can either Win $3, Win $1, Lose $1, or, in some sporadic cases, actually Lose $3.
g <- ggplot(three_bet_graph, aes(x = reorder(win_loss_bucket, win_loss_perc), y = win_loss_perc)) + geom_bar(stat = "identity", fill = "#a0acb8") + geom_text(aes(x = win_loss_bucket, y = win_loss_perc + 0.02, label = scales::percent(win_loss_perc))) + coord_flip() + theme_classic() + theme(axis.text.x = element_blank()) + xlab(label = "") + ylab(label = "") + labs(title = "Betting Outcome Summary", subtitle = "3 Bets for 1,000 Experiments") g
Since each experiment only includes 3 bets, the odds of you paying out money to your friend is 10.2% despite having the odds heavily in your favor. There is even a 0.9% chance that you might owe your friend $3 after 3 bets.
Conversely, you came out ahead about 89.8% of the time. You won three dollars 50.3% of the time and won one dollar 39.5% of the time.
Q4: What is the minimum number of bets needed to have a ~98% chance of a net win given 4:1 odds?
How many bets in a row would it take to increase your probability of a net win to 98%? Based on the odds presented earlier, the number of consecutive bets you would need to have a 98% chance of coming out ahead is ~9 bets.
Let’s set up various betting scenarios and visualize the probabilities that will help us prove this out. We will simulate 1 bet, 3 bet, 5 bet, 7 bet, and 9 bet scenarios using 1,000 experiments for each scenario and count how many times you owe your friend in each experiment. We will then tally and graph the results.
set.seed(123) bet_scenario_tbl <- rbind( str_c(rep(c(1:1), each = 1000), " Bet") %>% enframe(), str_c(rep(c(3:3), each = 3000), " Bet") %>% enframe(), str_c(rep(c(5:5), each = 5000), " Bet") %>% enframe(), str_c(rep(c(7:7), each = 7000), " Bet") %>% enframe(), str_c(rep(c(9:9), each = 9000), " Bet") %>% enframe() ) %>% rename(bet_scenario = value) experiment_num_tbl <- rbind( rep(c(1:1000), each = 1) %>% enframe(), rep(c(1:1000), each = 3) %>% enframe(), rep(c(1:1000), each = 5) %>% enframe(), rep(c(1:1000), each = 7) %>% enframe(), rep(c(1:1000), each = 9) %>% enframe() ) %>% rename(experiment_num = value) random_value_tbl <- runif(25000, 0, 1) %>% enframe() %>% rename(random_value = value) mult_bet_scenario_tbl <- cbind(bet_scenario_tbl, experiment_num_tbl, random_value_tbl) %>% select(bet_scenario, experiment_num, random_value) head(mult_bet_scenario_tbl)
Now that the data is set up let’s add the same calculated fields as we did earlier.
mult_bet_scenario_tbl <- mult_bet_scenario_tbl %>% mutate(win_loss = case_when(random_value <= 0.20 ~ -1, TRUE ~ 1))
Now we can tally up the results of each of the 1,000 experiments for each betting scenario.
mult_bet_scenario_aggr <- mult_bet_scenario_tbl %>% group_by(bet_scenario, experiment_num) %>% summarise(win_loss_sum = sum(win_loss)) %>% ungroup() %>% mutate(win_loss_bucket = case_when(win_loss_sum < 0 ~ "Number of Times You Owe", TRUE ~ "Number of Times You Collect")) %>% mutate(win_loss_tally = 1) mult_bet_scenario_aggr %>% glimpse()
To set up the bar charts, let’s aggregate the results.
mult_bet_scenario_graph <- mult_bet_scenario_aggr %>% select(bet_scenario, win_loss_bucket, win_loss_tally) %>% group_by(bet_scenario, win_loss_bucket) %>% summarise(win_loss_tally = sum(win_loss_tally)) %>% ungroup() %>% mutate(win_loss_perc = win_loss_tally / 1000) head(mult_bet_scenario_graph)
Next, we can visualize multiple betting scenarios played out over 1,000 experiments each. In other words, how do your chances of not having to pay increase with the number of consecutive bets made?
g <- ggplot(mult_bet_scenario_graph, aes(x = reorder(win_loss_bucket, win_loss_perc), y = win_loss_perc)) + geom_bar(stat = "identity", fill = "#a0acb8") + geom_text(aes(x = win_loss_bucket, y = win_loss_perc + 0.2, label = scales::percent(win_loss_perc))) + coord_flip() + theme_classic() + theme(axis.text.x = element_blank()) + xlab(label = "") + ylab(label = "") + labs(title = "Betting Outcome Summary", subtitle = "Based on 1,000 Simulations for Each Scenario") + ylim(-0.2, 1.4) + facet_wrap( ~ bet_scenario, ncol = 3) g
Using the probabilities we’ve assumed throughout this article, if you decided to bet your friend that the next person is indeed a strawberry fan, there is a 19.8% chance that you would lose that bet. It is not surprising that this value is almost identical to the original ~20.0% probability of someone preferring vanilla.
However, as you increase the number of bets, the chances of you losing drop reasonably quickly. To summarize:
- The odds of you owing money is ~19.8% when 1 bet is made.
- The odds of you owing money is ~11.2% when 3 bets are made.
- The odds of you owing money is ~6.9% when 5 bets are made.
- The odds of you owing money is ~4.0% when 7 bets are made.
- The odds of you owing money is ~1.4% when 9 bets are made.
Note that the numbers above can vary slightly depending on how many experiments are run but are reasonable estimates of how the odds change as the number of bets increases.
Conclusion & Next Steps
To summarize, it is essential to understand the probability and value assigned to each outcome and how many times you get to repeat an event when calculating Expected Values.
When you increase the number of bets or events, you can increase your chances of a net win if the odds are in your favor. That is why the casinos are so successful even when they only have a slight advantage. The casinos have very thin advantages over the players. However, casinos can bet thousands of times, dramatically increasing their chances of having an overall net gain. The phrase “the house always wins” is a very true statement.
