Cracking the Code
What is another term for the mean of a discrete probability distribution?
Expected Value or Average
What notation or representation is used for the expected value?
E(X)
What specific real-world application of expected value deals with profits, losses, and risks?
business
What symbol represents the mean of a discrete probability distribution?
"mu" or "μ"
TRUE or FALSE: The mean of a discrete probability distribution describes the central location of the data or outcomes.
TRUE
TRUE or FALSE: The expected value is a weighted average and may not necessarily be one of the actual values in the data set. It represents the long-term average outcome over many trials, but it does not have to be a possible result itself.
TRUE
How is the mean of a discrete probability distribution different from the arithmetic mean of a data set?
The mean of a discrete probability distribution is a weighted average, where each value is multiplied by its probability before summing it up.
A random variable X can take the values 1, 2, or 3 with probabilities 0.3, 0.5, and 0.2, respectively. Compute the mean of this distribution.
1.9
Why do casinos and lotteries use expected value when designing their games?
Casinos and lotteries use expected value to ensure they make a profit in the long run. The expected value for players is usually negative, meaning they are more likely to lose money over time, while the casino or lottery company benefits.
What is the formula for computing the mean of a discrete probability distribution?
∑ [X∙P(X)] or "the summation of X times P(X)"
A random variable X can take the values 3, 4, or 5 with probabilities 0.1, 0.5, and 0.4, respectively. Compute the mean.
4.3
A basketball player has a 40% chance of making a free throw. In a game, he takes 5 free throws. What is the expected number of free throws he will make?
2
What is the primary step in computing the mean of a discrete probability distribution?
Multiply each value of the random variable by its corresponding probability, then sum all the products.
A random variable X takes the values 1, 2, and one unknown value with probabilities 0.2, 0.6, and 0.2, respectively. With the mean, which is 2.4, find the unknown value.
5
Imagine you are an entrepreneur starting a new business. How could you use the concept of expected value to help you plan for success?
An entrepreneur can use expected value to estimate profits and losses, assess investment risks, and determine pricing strategies. By analyzing the expected revenue from different business strategies, they can choose the option that maximizes their potential earnings while minimizing risks.