The ACT Math section often features probability and statistics questions that surpass fundamental dataset metrics such as mean, median, mode, and range. Understanding how to calculate the expected value of a random discrete variable is a crucial skill for tackling these problems. In this blog post, we will break down the concept of expected value and provide step-by-step instructions on how to solve these problems on the ACT Math section.
What Is Expected Value?
Expected value, sometimes referred to as the mean or average value, is a fundamental concept in probability and statistics. It quantifies the long-term average of a random variable. In the context of the ACT Math section, you’ll typically encounter discrete random variables, which take on specific, distinct values.
The formula for calculating the expected value of a discrete random variable, often denoted as E(X), is:
E(X) = Σ [x * P(X=x)]
Where:
E(X) represents the expected value.
x represents the possible values of the random variable.
P(X=x) represents the probability of each value x occurring.
The Greek letter Σ denotes the sum of the values.
Step-by-Step Guide to Calculate Expected Value
Now, let’s dive into a step-by-step guide for calculating the expected value of a discrete random variable:
- Identify the random variable and its possible values: The first step is to determine the random variable in the problem and list its possible values. This may be explicitly stated in the question or can be inferred.
- Determine the probabilities: You need to find the probability of each possible value of the random variable occurring. This information may be given directly in the problem or can be calculated based on the information provided.
- Multiply each value by its probability: For each value of the random variable, multiply the value by its corresponding probability. This means you will have a series of products (x * P(X=x)) for each possible value of x.
- Sum the products: Add up all the products from step 3. This will give you the expected value (E(X)).
Let’s work through an example:
You are rolling a fair six-sided die, and you want to find the expected value of the outcome.
1. Identify the random variable and its possible values:
Random variable X: The outcome of rolling a fair six-sided die.
Possible values of X: {1, 2, 3, 4, 5, 6}
2. Determine the probabilities:
Since it’s a fair die, each outcome has a 1/6 probability (P(X=1) = P(X=2) = P(X=3) = P(X=4) = P(X=5) = P(X=6) = 1/6).
3. Multiply each value by its probability:
For each value x, calculate x * P(X=x):
E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6)
4. Sum the products:
E(X) = (1/6) + (2/6) + (3/6) + (4/6) + (5/6) + (6/6)
E(X) = (21/6)
Simplify the fraction:
E(X) = 7/2
Therefore, the expected value of the outcome when rolling a fair six-sided die is 7/2, which is equal to 3.5.
Conclusion
Calculating the expected value of a random discrete variable is a valuable skill for the ACT Math section. By understanding the concept and following the step-by-step guide provided in this blog post, you can confidently approach probability and statistics questions involving expected values. Remember to practice a variety of problems to solidify your understanding and perform well on the test.
Mindfish can tailor a Test Prep Tutoring program specifically to your needs and pair you with a tutor who can help you identify challenging or unfamiliar questions on the test (like expected value) and guide you toward solutions. Reach out to our Admin team at 720-204-1041 or
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