var outcomePairCount = 3;
function addOutcomePair() {
outcomePairCount++;
var container = document.getElementById('outcomePairsContainer');
var newPairDiv = document.createElement('div');
newPairDiv.className = 'outcome-pair';
newPairDiv.id = 'outcomePair_' + outcomePairCount;
newPairDiv.innerHTML = `
`;
container.appendChild(newPairDiv);
}
function calculateExpectedValue() {
var totalExpectedValue = 0;
var totalProbability = 0;
var resultDiv = document.getElementById('expectedValueResult');
resultDiv.innerHTML = ";
for (var i = 1; i <= outcomePairCount; i++) {
var outcomeValueInput = document.getElementById('outcomeValue_' + i);
var probabilityInput = document.getElementById('probability_' + i);
if (!outcomeValueInput || !probabilityInput) {
continue; // Skip if elements don't exist (e.g., after removing a row)
}
var outcomeValue = parseFloat(outcomeValueInput.value);
var probability = parseFloat(probabilityInput.value);
if (isNaN(outcomeValue) || isNaN(probability)) {
resultDiv.innerHTML = 'Please enter valid numbers for all outcome values and probabilities.';
return;
}
if (probability 1) {
resultDiv.innerHTML = 'Probabilities must be between 0 and 1.';
return;
}
totalExpectedValue += outcomeValue * probability;
totalProbability += probability;
}
// Check if total probability is close to 1 (allowing for floating point inaccuracies)
if (Math.abs(totalProbability – 1) > 0.001 && totalProbability > 0) {
resultDiv.innerHTML = 'Warning: The sum of probabilities is ' + totalProbability.toFixed(3) + ', which is not exactly 1.0. This might indicate missing outcomes or incorrect probabilities.';
} else if (totalProbability === 0) {
resultDiv.innerHTML = 'Please enter at least one outcome with a non-zero probability.';
return;
}
resultDiv.innerHTML += 'The
';
}
.calculator-container {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: #f9f9f9;
border: 1px solid #ddd;
border-radius: 8px;
padding: 25px;
max-width: 600px;
margin: 30px auto;
box-shadow: 0 4px 12px rgba(0, 0, 0, 0.08);
}
.calculator-container h2 {
color: #333;
text-align: center;
margin-bottom: 25px;
font-size: 1.8em;
}
.outcome-pair {
display: grid;
grid-template-columns: 1fr 1fr;
gap: 15px;
margin-bottom: 15px;
padding: 15px;
background-color: #fff;
border: 1px solid #eee;
border-radius: 5px;
}
.outcome-pair label {
font-weight: bold;
color: #555;
align-self: center;
}
.outcome-pair input[type="number"] {
width: 100%;
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
box-sizing: border-box;
font-size: 1em;
}
button {
background-color: #007bff;
color: white;
padding: 12px 20px;
border: none;
border-radius: 5px;
cursor: pointer;
font-size: 1.1em;
margin-top: 15px;
width: auto;
display: inline-block;
margin-right: 10px;
transition: background-color 0.3s ease;
}
button:hover {
background-color: #0056b3;
}
#addOutcomeBtn {
background-color: #28a745;
}
#addOutcomeBtn:hover {
background-color: #218838;
}
.calculator-result {
margin-top: 25px;
padding: 15px;
background-color: #e9f7ef;
border: 1px solid #d4edda;
border-radius: 5px;
font-size: 1.2em;
color: #155724;
text-align: center;
}
.calculator-result p {
margin: 0;
}
.calculator-result .error {
color: #721c24;
background-color: #f8d7da;
border-color: #f5c6cb;
padding: 10px;
border-radius: 4px;
}
.calculator-result .warning {
color: #856404;
background-color: #fff3cd;
border-color: #ffeeba;
padding: 10px;
border-radius: 4px;
}
/* Article Styling */
.article-content {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
line-height: 1.7;
color: #333;
max-width: 800px;
margin: 40px auto;
padding: 0 15px;
}
.article-content h3 {
color: #007bff;
margin-top: 30px;
margin-bottom: 15px;
font-size: 1.6em;
}
.article-content p {
margin-bottom: 1em;
text-align: justify;
}
.article-content ul {
list-style-type: disc;
margin-left: 20px;
margin-bottom: 1em;
}
.article-content li {
margin-bottom: 0.5em;
}
.article-content strong {
color: #0056b3;
}
Understanding Expected Value
The Expected Value (EV) is a fundamental concept in probability theory and statistics, representing the average outcome of a random variable over a large number of trials. It's a powerful tool for decision-making under uncertainty, helping individuals and organizations quantify the potential long-term gain or loss associated with a particular choice or event.
Unlike a simple average, expected value takes into account the probability of each possible outcome. This means that outcomes that are more likely to occur contribute more to the expected value than less likely ones.
The Formula for Expected Value
The calculation for expected value is straightforward. If you have a set of possible outcomes, each with its own value and probability of occurring, the expected value is the sum of the products of each outcome's value and its probability.
Mathematically, it's expressed as:
EV = (Outcome1 × Probability1) + (Outcome2 × Probability2) + … + (Outcomen × Probabilityn)
Where:
- Outcomei is the value of the i-th possible outcome.
- Probabilityi is the probability of the i-th outcome occurring.
- The sum of all probabilities (Probability1 + Probability2 + … + Probabilityn) must equal 1 (or 100%).
Practical Applications
Expected value is widely used across various fields:
- Gambling and Games of Chance: It helps determine if a game is fair or if it favors the player or the house in the long run. A positive EV suggests a long-term gain for the player, while a negative EV indicates a long-term loss.
- Business and Finance: Companies use EV to evaluate potential investments, project revenues, assess risks, and make strategic decisions. For example, a company might calculate the EV of launching a new product by considering different sales scenarios and their probabilities.
- Insurance: Actuaries use expected value to set premiums, ensuring that the company collects enough money to cover anticipated claims and make a profit.
- Decision Theory: It provides a quantitative basis for choosing between different courses of action when the outcomes are uncertain.
Example Scenario
Consider a simple game where you pay $5 to play. You roll a standard six-sided die:
- If you roll a 6, you win $20.
- If you roll a 5, you win $10.
- If you roll a 1, 2, 3, or 4, you win nothing.
Let's calculate the expected value of playing this game. First, we define the net outcomes and their probabilities:
- Outcome 1 (Roll a 6): Net gain = $20 (win) – $5 (cost) = $15. Probability = 1/6.
- Outcome 2 (Roll a 5): Net gain = $10 (win) – $5 (cost) = $5. Probability = 1/6.
- Outcome 3 (Roll 1, 2, 3, or 4): Net gain = $0 (win) – $5 (cost) = -$5. Probability = 4/6.
Now, apply the expected value formula:
EV = ($15 × 1/6) + ($5 × 1/6) + (-$5 × 4/6)
EV = ($2.50) + ($0.83) + (-$3.33)
EV = $0.00
In this specific example, the expected value is $0.00. This means that, on average, if you play this game many times, you would expect to break even. If the EV were positive, it would be a favorable game; if negative, an unfavorable one.
Use the calculator above to input your own outcome values and probabilities to quickly determine the expected value for your specific scenarios.