Expected Values, Main Ideas!!!

StatQuest with Josh Starmer
28 Mar 202113:39
EducationalLearning
32 Likes 10 Comments

TLDRIn this StatQuest episode, host Josh Starmer explores the concept of expected values with a whimsical visit to 'Statland.' He introduces a hypothetical scenario involving a bet on whether the next person encountered has heard of the movie 'Troll 2.' Using data from Statland, he calculates the probabilities and expected values associated with the bet, demonstrating through mathematical calculations that even with a high chance of winning, the average outcome can still be profitable over many iterations. The video simplifies complex statistical concepts, making them accessible and engaging for viewers interested in statistics and probability.

Takeaways
  • 🎬 The video is about explaining the concept of expected values in a clear and engaging manner.
  • 🗺️ The story takes place in a fictional place called 'Statland' where two types of people exist: those who have heard of the movie 'Troll 2' and those who haven't.
  • 👥 The population of Statland consists of 213 people, with 37 having heard of 'Troll 2' and 176 who haven't.
  • 📊 The probabilities of meeting someone who has or hasn't heard of 'Troll 2' are calculated to be 0.17 and 0.83, respectively.
  • 💰 The concept of betting and expected value is introduced with a hypothetical bet on whether the next person met has heard of 'Troll 2'.
  • ✅ The expected value of the initial bet is calculated to be a gain of 66 cents per bet on average, indicating it would be profitable to accept the bet.
  • 🔢 The expected value is derived from multiplying each outcome by its probability and summing these products, which simplifies to 0.66 cents in this scenario.
  • 🤔 The video discusses the idea of making the bet multiple times (100 times) to predict overall gains or losses, which is a common approach in understanding expected values.
  • 💡 The video also explores a modified bet scenario where the outcomes and their probabilities change, resulting in a different expected value.
  • 📚 The script mentions that while the video focuses on discrete events, there will be future content on continuous events and their expected values.
  • 📈 The video concludes with a call to action for viewers to subscribe, support the channel, and check out additional resources for learning more about statistics.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is explaining the concept of expected values in statistics.

  • Who is the host of the video?

    -Josh Starmer is the host of the video.

  • What is the fictional place introduced in the video to explain statistics?

    -The fictional place introduced in the video to explain statistics is called Statland.

  • What movie is used as an example in the video to explain expected values?

    -The movie 'Troll 2' is used as an example to explain expected values.

  • How many people in Statland have heard of the movie 'Troll 2'?

    -37 people in Statland have heard of the movie 'Troll 2'.

  • What is the total population of Statland according to the video?

    -The total population of Statland is 213 people.

  • What is the probability that a randomly selected person in Statland has heard of 'Troll 2'?

    -The probability that a randomly selected person in Statland has heard of 'Troll 2' is 0.17.

  • What is the expected value of the bet if Statsquatch bets one dollar that the next person met has heard of 'Troll 2'?

    -The expected value of the bet is 66 cents, meaning on average, you would expect to gain 66 cents each time you make the bet.

  • How many times can the bet be made according to the video?

    -The bet can be made 100 times according to the video.

  • What is the expected value if Statsquatch changes the bet to pay 10 dollars if the next person has heard of 'Troll 2' and asks for 1 dollar if they haven't?

    -The expected value of the new bet is 87 cents, indicating an average gain of 87 cents per bet.

  • What is the Statquest study guide mentioned in the video for?

    -The Statquest study guide is for reviewing statistics and machine learning offline.

  • How can viewers support Statquest according to the video?

    -Viewers can support Statquest by subscribing, contributing to the Patreon campaign, becoming a channel member, buying original songs or merchandise, or making a donation.

Outlines
00:00
🎲 Introduction to Expected Values with a Troll 2 Bet

In this first paragraph, the video introduces the concept of expected values using a hypothetical scenario in a place called Statland. The host, Josh Starmer, sets up a bet with a character named Statsquatch, who believes the next person they meet will have heard of the movie 'Troll 2'. The data reveals that out of 213 people in Statland, 37 have heard of the movie, which is considered a bad film, implying it's unexpected knowledge. The probabilities are calculated: 0.17 chance someone has heard of 'Troll 2' and 0.83 chance they haven't. The host then uses these probabilities to evaluate the bet's outcome, suggesting that despite the low probability of winning, the potential loss is minimal, and the high chance of winning makes it an attractive bet.

05:01
📊 Calculating Expected Value Over Multiple Bets

This paragraph delves into the idea of making the 'Troll 2' bet multiple times to understand the expected value. By multiplying the probabilities of winning and losing by the number of bets (100), the host calculates the expected outcomes. With a 17% chance of losing, the expected loss is about 17 times per 100 bets, amounting to a loss of $17. Conversely, with an 83% chance of winning, the expected winnings are about 83 times per 100 bets, totaling $83. By combining these, the host finds that over 100 bets, the expected gain is approximately $66, which simplifies to an average expected value of 66 cents per bet. This demonstrates that even though there's a risk of losing individual bets, the long-term average is a gain, illustrating the power of expected value in decision-making.

10:01
🧮 Adjusting Bets and Recalculating Expected Values

In the third paragraph, the scenario changes as Statsquatch modifies the bet's terms, offering to pay $10 if the next person has heard of 'Troll 2' and requiring a $1 payment if they haven't. The host then recalculates the expected value with these new outcomes. The expected value is calculated by multiplying each outcome by its probability and summing the results. With the new bet, the expected value is 0.87, indicating an average gain of 87 cents per bet. This suggests that with the adjusted terms, the bet becomes more favorable for the host, and Statsquatch would be at a disadvantage in the long run, highlighting the importance of understanding expected values when making decisions, especially in gambling scenarios.

Mindmap
Keywords
💡Expected Value
Expected value is a fundamental concept in probability and statistics that represents the average outcome of a random event if it were repeated a large number of times. In the video, it is used to determine the long-term average gain or loss from a bet. The script illustrates this by calculating the expected value of a bet on whether someone in 'Statland' has heard of the movie 'Troll 2', showing that even though there is a chance of losing, the expected value is a gain of 66 cents per bet.
💡Statland
Statland is a fictional place created for the purpose of the video to illustrate statistical concepts. It is a simplified world where people are divided into two groups based on whether they have heard of the movie 'Troll 2' or not. The script uses Statland to demonstrate how probabilities and expected values can be calculated and applied to a hypothetical scenario.
💡Probability
Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In the script, probabilities are calculated to determine the chance that a randomly selected person from Statland has heard of 'Troll 2', which is 0.17, and the chance that they have not, which is 0.83. These probabilities are crucial for understanding and calculating the expected value of the bet.
💡Bet
A bet is a wager on an uncertain outcome, often involving the possibility of winning or losing money. In the video, the concept of a bet is used as a practical example to explain expected values. The script discusses two different bets involving knowledge of the movie 'Troll 2', and how the expected value of these bets can influence the decision to accept or decline them.
💡Troll 2
Troll 2 is a 1990 movie that is often referenced in the script as an example for explaining statistical concepts. It is used as a known or unknown entity to divide the population of Statland into two groups, which simplifies the demonstration of how probabilities and expected values are calculated.
💡Statistics
Statistics is the discipline that concerns the collection, analysis, interpretation, presentation, and organization of data. The video uses the context of a bet to explain various statistical concepts such as expected value and probability. It shows how these concepts can be applied to make informed decisions about uncertain events.
💡Discrete Events
Discrete events are outcomes that can be counted and are distinct from one another, as opposed to continuous events that can take on a range of values. In the script, the knowledge of 'Troll 2' is a discrete event because a person either has heard of it or has not, with no in-between states. The video explains how to calculate expected values for such discrete events.
💡Continuous Events
Continuous events are those that can take on any value within a range, such as the time between text messages. Although not the main focus of the video, the script mentions that in a future 'StatQuest', they will discuss how to calculate expected values for continuous events, indicating the broader applicability of statistical methods.
💡Sample Variance
Sample variance is a measure that provides information about the dispersion of a set of data points in a sample. While not the central topic of the video, the script mentions dividing the sample variance by 'n-1' as part of a process to be explained in future content, hinting at more advanced statistical concepts.
💡StatQuest
StatQuest is the name of the series and also the channel that produces the video content. It is focused on explaining statistics and machine learning concepts in an accessible way. The script uses StatQuest as a brand to promote related study guides and to encourage viewers to subscribe for more content.
Highlights

Introduction to expected values and their clear explanation.

Visit to the fictional place called Statland to illustrate the concept.

Bet between the host and Statsquatch on whether the next person has heard of the movie 'Troll 2'.

Data collection on the awareness of 'Troll 2' among Statland's population.

Calculation of probabilities for people in Statland having heard or not heard of 'Troll 2'.

Decision-making on accepting a bet based on calculated probabilities.

Simulating 100 bets to predict outcomes and expected monetary gains or losses.

Calculation of expected value per bet and its interpretation.

Use of statistical notation to express the expected value of the bet.

Explanation of how the expected value represents the long-term outcome of repeated bets.

Calculation of expected value for a modified bet with different outcomes and probabilities.

Discussion on the difference in expected value when the bet's terms change.

Introduction of sigma notation for calculating the expected value.

Illustration of how the number of bets (100) cancels out in the calculation, simplifying the expected value formula.

Explanation of the implications of a positive expected value in the context of repeated betting.

Promotion of StatQuest study guides for offline review of statistics and machine learning.

Invitation to subscribe for more StatQuest content.

Options provided for supporting StatQuest through Patreon, channel membership, merchandise, or direct donations.

Transcripts

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