91Ó°ÊÓ

The Expected Value, often represented by the symbol \(E(X)\), is a key concept in probability that reflects the average outcome if an experiment is repeated many times. In the context of a binomial distribution, which deals with a fixed number of trials and only two possible outcomes per trial, the expected value indicates the average number of successes you can anticipate. To find the expected value in a binomial experiment, you use the formula: \[ E(X) = n \cdot p \] where \(n\) is the total number of trials, and \(p\) is the probability of success on an individual trial. For instance, in our exercise with \(n = 8\) trials and a success probability of \(p = 0.20\), the expected value is calculated as follows: \[ E(X) = 8 \cdot 0.20 = 1.6 \] This means that, on average, we expect to see 1.6 successes out of 8 trials. It's important to note that while we can't have a fractional success in reality, this value represents a theoretical average typically rounded in practical terms or interpreted over many repetitions.

The Standard Deviation, denoted by \(\sigma\), is a measure of how much variation or dispersion there is from the expected value in a probability distribution. In simpler terms, it tells us how much the values tend to spread out from the expected value.In a binomial distribution, the formula to compute the standard deviation is: \[ \sigma = \sqrt{n \cdot p \cdot (1-p)} \] Here, \(n\) signifies the number of trials and \(p\) the probability of success on an individual trial. In our specific case with \(n = 8\) and \(p = 0.20\): \[ \sigma = \sqrt{8 \cdot 0.20 \cdot (1-0.20)} = \sqrt{8 \cdot 0.20 \cdot 0.80} = \sqrt{1.28} \approx 1.13 \] The standard deviation of approximately 1.13 suggests that the number of successes typically varies by about 1.13 around the expected value of 1.6. A lower standard deviation would indicate that the actual results are closer to the expected value, whereas a higher standard deviation implies more variability.

The Binomial Probability Table is a helpful tool in statistics that provides probabilities for different numbers of successes in a binomial experiment. By using this table, we can quickly determine the likelihood of obtaining a specific number of successes without doing complex calculations for each possible outcome.In our exercise, the binomial probability table is used to decide if getting 5 or more successes in 8 trials with a success probability of 0.20 is considered unusual. To solve this:

• Look up the probabilities for each possible outcome of interest: \( P(X = 5) \), \( P(X = 6) \), \( P(X = 7) \), and \( P(X = 8) \).

These individual probabilities are added to discover the overall likelihood of having at least 5 successes. This cumulative probability can be found through calculations or by referring to statistical software or binomial probability tables. In our case, the cumulative probability \( P(X \geq 5) \) was approximately 0.032, a value below 0.05, indicating it is indeed unusual to achieve 5 or more successes under the given conditions. This highlights how valuable a binomial probability table can be in quickly assessing probabilities in statistical problems.