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The mean of a probability distribution, often called the expected value, gives us a sense of the average or the central tendency. It tells us what result we can expect on average if we were to repeat the experiment many times. In this context, the mean represents the average number of correct matches guessed.To compute the mean, we multiply each possible outcome by its probability and then sum these products. For our distribution:- Outcome 0 has probability 0.333- Outcome 1 has probability 0.5- Outcome 2 has probability 0- Outcome 3 has probability 0.167Thus, the mean \( \mu \) is calculated as:\[ \mu = (0 \times 0.333) + (1 \times 0.5) + (2 \times 0) + (3 \times 0.167) \]Breaking it down:- \( 0 \times 0.333 = 0 \)- \( 1 \times 0.5 = 0.5 \)- \( 2 \times 0 = 0 \)- \( 3 \times 0.167 = 0.501 \)Adding these gives \( \mu = 0 + 0.5 + 0 + 0.501 = 1.001 \). So, on average, you can expect to guess 1 correct match out of the three.

Variance is an essential measure in statistics that tells us how much the values in a distribution spread out from the mean. It gives the expected squared distance of each outcome from the mean, providing a clearer picture of actual variability.To calculate the variance \( \sigma^2 \), follow these steps:1. Compute the squared difference between each outcome and the mean.2. Multiply each squared difference by the corresponding probability.3. Sum all these values.For our distribution, the outcomes are compared with the mean \( \mu = 1.001 \):- For 0 correct: \( (0 - 1.001)^2 \times 0.333 \)- For 1 correct: \( (1 - 1.001)^2 \times 0.5 \)- For 2 correct: \( (2 - 1.001)^2 \times 0 \)- For 3 correct: \( (3 - 1.001)^2 \times 0.167 \)Calculating these:- \( 1.002001 \times 0.333 = 0.333364333 \)- \( 0.000001 \times 0.5 = 0.0000005 \)- \( 3.996001 \times 0 = 0 \)- \( 3.998001 \times 0.167 = 0.667668167 \)The variance \( \sigma^2 \) thus becomes:\[ 0.333364333 + 0.0000005 + 0 + 0.667668167 = 1.001033 \]

The standard deviation is a widely used measure of the amount of variation or dispersion within a distribution. While variance gives us information about spread, the standard deviation provides that information in more understandable units, since it corresponds to the scale of measurement.To find the standard deviation \( \sigma \), simply take the square root of the variance:\[ \sigma = \sqrt{1.001033} \approx 1.000516 \]The standard deviation of about 1.000516 suggests that the number of correct matches you can guess tends to deviate from the mean by about one match. This insight can help understand the consistency of guesses. A smaller standard deviation would indicate guesses are closely concentrated around the mean, while a larger one shows more spread.