After watching this video lesson, you will understand what expected value means. You will learn how to use this value to decide whether a certain action is worth taking or not.
In Las Vegas, Nevada, there are a lot of casinos where you can play games for a bit of money. Each time you play these games, you can expect to win nothing, something, or a lot of money back! Of course, if you hit the jackpot and win a million dollars from just one game, you would think that these games are great! But keep playing and you will find yourself losing more and more money. These casinos are a business, and they are in it for the money just like any other business. So all the games that they offer have been mathematically and statistically scrutinized by the business team to make sure that the casino ends up making money.
How do they do this? They calculate what is called the expected value of the game. The expected value is the average value you can expect after a large number of rounds. This means that the more and more games you play in a casino, the closer you will get to the expected value for your earnings, or lack thereof. You can use this value to determine whether a game is worth playing or not. If the expected value of a game is -$5.00, for example, you might say it's not worth losing $5 to play it, or you might say that the fun of playing the game is worth losing $5.
Because this expected value is an average, you can expect to hit this number when playing the game. For example, a game of dice may have an expected value of $2, but the only choices are to win $0, $5, or $10. You will never win $2, but if you play enough rounds, you will find that your earnings per game will get closer and closer to $2.
Probability in Each Event
To calculate the expected value of a particular game, the casinos need to know the probability of each event that may happen in the game. For example, in a game of roulette, they need to know the probability or chance of the ball landing in each of the numbers on the roulette wheel. In a dice game, they need to know the probability of the dice landing on each of its numbers 1 through 6. Then, to calculate the expected value, they multiply each event with its probability and sum it all up. This gives them the expected value of that particular game.
We can write this calculation in formula form by using the summation symbol like this:
In this formula, x represents our event. So this formula is telling us to multiply our event with its probability, and then sum all these multiplications together. For example, if the probability of rolling a 5 in a dice game and winning $10 is 1/6, we would multiply the $10 (the event) with 1/6 (the probability of it happening), and then add it to all our other choices.
In a game of dice, since there are only six possible events, we only have to add up 6 events. For the roulette wheel, since there are 38 choices, we will need to add up 38 events.
Let's look at an example:
It costs $3 to play a game of dice. You can expect to win $5 if you roll a 5 and $10 if you roll a 6. What is the expected value of this game? This is in addition to winning your $3 back, so a net win of $5 or $10.
We first need to make a table of our events and the probability of it happening. We know that the probability of rolling each number on the dice is 1/6 for each number. So we can write a table listing our dice numbers, the money we earn when we roll that number, and the probability of it happening.
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We have six choices when rolling a dice. I've listed them with the cost of rolling each and its related probability. If you roll a 1, you lose $3 because that's the amount you paid to play the game. Now we can begin to calculate our expected value. Our x (our event) is our earnings. We can go through our table and multiply our earnings with the probability for each row, and then add them all up.
We have our expected value = (-3)(1/6) + (-3)(1/6) + (-3)(1/6) + (-3)(1/6) + (5)(1/6) + (10)(1/6). This adds up to (-0.5) + (-0.5) + (-0.5) + (-0.5) + (0.833) + (1.667) = 0.5. Our expected value is $0.50 per game.
This means that if we keep playing more and more games, we can expect to make 50 cents per game. Looking at this expected value, you might say that your time is worth more than 50 cents per game and decide that the game is not worth it, or you might say 'hey, it's worth it because you've got nothing better to do. Might as well make a bit of money.'
If you were the casino, on the other hand, you would see this expected value and say that we can't offer this game at this price because we will lose money. So, what do you do? You can increase the cost per game or decrease the possible earnings. You calculate the expected value again until you get a value that means you make money. If the expected value is negative, then the player loses money and the casino gains money.
Of course, we can have situations where the probability of each event is different. In this case, you follow the same format as we did for the dice game (multiplying the event with its probability and summing them all up), but instead of all the probabilities being 1/6, they will each be different.
Now that you know the process, you can apply this to business decisions. As a business owner, a product maker might come to you and ask you to sell her items. She has a certain track record and you can see the probability of her items selling at a particular price point. You can use your newly-learned skills to calculate whether it would be worth it for you to sell her items at a particular price point.
Let's review what we've learned:
The expected value is an average value you can expect after a large number of rounds. For example, if the expected value of playing a game is -$1, you can expect to lose a dollar each game as you keep playing more and more games, even if your possible wins are only $0 and $10.
To calculate this value, you multiply each event with its probability and add them all up. The formula you can use for this can be written with the summation symbol:
After you've reviewed this video lesson, you should be able to:
Define expected value
Identify the formula to calculate expected value
Explain the real-world implications of finding expected value
Expected Value in Probability - A Practical Exercise:
The following exercise is designed to help students apply their knowledge of the expected value in a real-life context.
You just received an inheritance of $100,000 and you want to invest it as part of your retirement savings. Naturally, every investment option has a different level of risk and a different return on investment. Your financial advisor has found two mutual funds that meet your risk profile and investment criteria. The information on both funds is given below.
Return in 1 year ($)
Return in 1 year ($)
When presented with this information, you tell the financial advisor that Fund B is obviously the best choice... "The lowest possible return is the same but Fund B can return $10,000!" He tells you to recall your knowledge from expected values in probability and to reconsider.
1. Determine the expected value of each fund's returns.
2. Based on #1, which fund would you prefer? Why?
Using the expected value formula, we will multiply each event with its probability and add them all up for each fund.
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