Expected Value | Objective vs Subjective | Statistics and Probability EP8 - YouTube

The expected value or mathematical expectation  is the average value of a random variable over  

a large number of experiments. I will explain all  you need to know about it in the next 5 minutes.  

If there is a share worth $100 at this moment, and  you know there is a 30% chance it will worth $120  

next month; 30% chance remains the same;  and 30% chance down to $70.  

In this case, we know all the probabilities,

but how can you decide whether this share worth buying or not?

The expected value  can be used to answer this question.

For this example, we need to know the expected  value of the anticipated gain or loss for that investment

in one month's time.  In statistic and probability analysis, the expected value is  

calculated by multiplying each of possible  outcomes by the likelihood each outcome will occur 

and then summarizing all of those values. Let's  find out the expected profit value of our example:

See? Although there's more chance  this share will go up next month  

the expected value is negative  which indicates you shouldn't buy it  

Now let's see an example of using expected value  to make a strategy: In basketball different points  

are awarded to players based on where they are  when they shoot the ball. Behind the 3 point  

line will awards 3 points, and inside the  3 point line only award 2 points. However  

the shooting average will be getting lower  if the player is far away from the basket.  

Since the ultimate object of basketball is to  score more points let's find out how by calculate  

the expected value. One player's chance of  score behind the three-point line is 40%  

inside the three-point line is 50% and slam dunk is  60% Then the expected value of each score type is

This means over the long term each time  shooting behind the three-point line  

or slam dunk will lead to 1.2 points while shooting  from inside the three-point line will only get 1 point.

From our analysis, this player should  try to shoot less inside the three-point line  

in order to score more efficiently. I believe now  you get the idea that the expected value is useful  

to examine the random variable's long-term value.  Speaking of "long-term" you probably remember the  

law of large numbers which convert the observed  frequency to probability it is a bridge connects  

randomness to certainty! Further the expected  value covers the probability to a certain number  

which enables us to do the analysis. To make the  expected value work, we have to assign a value to  

each result like the score point in the basketball  example. Have you ever wondered how to balance the  

character's skill in a computer game? They are  all different, but no one is too strong or weak. 

Easy:) The game designer assigns a number to the  character's each aspect such as attack damage and defense,  

and then calculate the expected value.  All characters should have a similar expected value. 

If the character is too strong, lower the attack  damage a little bit or if it's too weak adjust  

its defense a little higher. Assign a value is  not always as easy as for the game characters  

You must know the old saying that "one  man's trash is another man's treasure" 

Most of the time there's no objective value.  for example, if you ask me to play a game:  

toaster dice, 1 point I will get $1  2 point I will get $2 and so on. the game  

The game will take me about 30' (to finish)   the expected value is $3.5

Will I play it? (Or should I play it?)

If I am waiting for a bus  and looking to do something to kill the time

at that moment 30 second worth almost 0 for me

so I will say yes. But if I'm working on an  urgent project and really need to make every  

second count, in that case, 30 seconds worth much more  than $3.5 so I would say no to that offer.  

Three takeaways: first, the expected value is the  average value of a random variable over long term.

It quantifies the variable which gives us  an indicator of whether it was doing or not.

second, in order to calculate the expected value  we have to assign a value to the random variable

third, most of the time there is  no objective value for a random  

variable for the same variable we may assign  a different value during different situations.

I'm always here at your service. See you in the next one!