What is formula that I use for calculating expected values?:

Question

Consider the following game. Each player flips 2 coins. The player gets 2 points for each result of heads and 1 point for each result of tails. Calculate the expectedvalue of points earned in the game.  List Outcome, Probability ?

Answer 1

Hi John!

 

The expected value is the payoff you would expect to get if you played a game (here, the coin flip game) infinitely many times. This just means if we kept playing over and over, your average payoff would approach the expected value.

 

Expected value formula: E(X) = x₁p₁ + x₂p₂ + x₃p₃ + x₄p₄ + ... This can keep going until every possibility is covered.

 

Okay, so what do the variables mean? E(X) means "expected value," what we're looking for.

 

Each x, for example, x₁, represents a different possible outcome for the game -> here, outcome meaning how many points you would win from the coin flip round.

 

Each p, for example p₁, represents the probability of the outcome next to it.

 

For this problem, there are actually 3 possible outcomes, because you are flipping two coins:

Outcome 1: getting 2 heads -> payoff of 4 points -> x₁ = 4

Outcome 2: getting 2 tails -> payoff of 2 points -> x₂ = 2

Outcome 3: getting 1 head, 1 tail -> payoff of 3 points -> x₃ = 3

 

What is the likelihood of getting any of these outcomes? Each of them has a 1 in 3 chance, or 1/3, because there is an equal chance that when you flip two coins, you get any of these combinations (this is because there is equal probability of getting either heads or tails, so getting two heads or two tails is not any more likely that one head and one tail, though our superstitions might tell us otherwise).

 

Now, we know:

p₁ = 1/3

p₂ = 1/3

p₃ = 1/3

 

Now, we can plug in our numbers into our expected value formula and solve:

 

E(X) = x₁p₁ + x₂p₂ + x₃p₃

E(X) = 4(1/3) + 2(1/3) + 3(1/3)

E(X) = 4/3 + 2/3 + 3/3

E(X) = 9/3

E(X) = 3

 

Our expected value is 3 points.

 

Hope this is helpful!

 

 

 

 

 

Answer 2

The expected value is the sum of the products of all outcomes and the respective probabilities. In your example, the outcomes and the probabilities are listed below.

 

Outcome Probability

2             0.25   (tails, tails)

3             0.50   (tails, heads or heads, tails)

4             0.25   (heads, heads)

 

The expected value is 2*0.25+3*0.50+4*0.25 = 3