Expected Value Calculator | Formula and Examples

Our online expected value calculator is used in probability theory and statistics to compute the average outcome of an event that has various possible results using the formula E(X)=∑(xi⋅pi)E(X)=∑(xi​⋅pi​).

Expected Value Calculator

Scenario	Outcome 1	Probability 1	Outcome 2	Probability 2	Outcome 3	Probability 3	Expected Value
Coin Flip Game	Win $5	0.5	Lose $3	0.5	–	–	$1.00
Investment A	Gain $1000	0.3	Gain $500	0.5	Lose $200	0.2	$460.00
Weather Forecast	Sunny (Profit $100)	0.6	Rainy (Loss $50)	0.3	Cloudy (Profit $20)	0.1	$47.00
Product Launch	Success ($10,000)	0.4	Moderate ($5,000)	0.35	Failure (-$2,000)	0.25	$5,050.00
Lottery Ticket	Win $1,000,000	0.0000001	Win $1,000	0.0001	No win ($0)	0.9998999	$0.20

1. Coin Flip Game
   • Expected Value = (5 0.5) + (-3 0.5) = 2.5 – 1.5 = $1.00
   • This simple game has a positive expected value, suggesting it’s favorable to play in the long run.
2. Investment A
   • Expected Value = (1000 0.3) + (500 0.5) + (-200 * 0.2) = 300 + 250 – 40 = $460.00
   • This investment has a positive expected value, indicating it’s potentially a good investment choice.
3. Weather Forecast
   • Expected Value = (100 0.6) + (-50 0.3) + (20 * 0.1) = 60 – 15 + 2 = $47.00
   • This scenario, possibly related to an outdoor event, shows a positive expected value.
4. Product Launch
   • Expected Value = (10000 0.4) + (5000 0.35) + (-2000 * 0.25) = 4000 + 1750 – 500 = $5,050.00
   • The product launch has a high positive expected value, suggesting it’s a promising venture.
5. Lottery Ticket
   • Expected Value = (1000000 0.0000001) + (1000 0.0001) + (0 * 0.9998999) = 0.1 + 0.1 + 0 = $0.20
   • Despite the high potential payout, the extremely low probability of winning results in a very low expected value.

How is the expected value calculated?

The expected value is calculated by multiplying each possible outcome by its probability of occurrence and then summing up all these products. This process involves the following steps:

1. Identify all possible outcomes of an event
2. Determine the probability of each outcome
3. Multiply each outcome’s value by its probability
4. Sum up all the resulting products

This method provides a weighted average of all possible outcomes, taking into account both their likelihood and their respective values.

More Calculators : Mean Absolute Deviation Calculator – Shannon Entropy Calculator

Expected Value Calculation Formula

The formula for calculating expected value is:

E(X) = Σ (x_i * p_i)

Where:

• E(X) is the expected value of random variable X
• x_i represents each possible outcome
• p_i is the probability of each outcome
• Σ denotes the sum of all products

This formula can be applied to both discrete and continuous probability distributions, though the calculation method may vary slightly for continuous distributions.

Chi square expected value

The chi-square test is a statistical method used to determine if there is a significant difference between the expected frequencies and the observed frequencies in one or more categories. In this context, the expected value plays a crucial role.

For a chi-square test, the expected value for each category is calculated as:

E = (row total * column total) / grand total

Where:

• E is the expected value
• Row total is the sum of observations in a row
• Column total is the sum of observations in a column
• Grand total is the sum of all observations

The chi-square statistic is then computed by comparing these expected values with the observed values, helping researchers determine if there’s a significant association between two categorical variables.

What is an example of expected value?

Suppose you’re playing a game where you roll a six-sided die. If you roll an even number (2, 4, or 6), you win $10. If you roll an odd number (1, 3, or 5), you lose $5.

To calculate the expected value:

1. Identify outcomes:
   • Win $10 (even roll)
   • Lose $5 (odd roll)
2. Determine probabilities:
   • P(even) = 3/6 = 1/2
   • P(odd) = 3/6 = 1/2
3. Multiply outcomes by probabilities:
   • Even: $10 * 1/2 = $5
   • Odd: -$5 * 1/2 = -$2.50
4. Sum up the products: E(X) = $5 + (-$2.50) = $2.50

The expected value of this game is $2.50. This means that, on average, you can expect to gain $2.50 per roll if you play this game many times.