You are asked to match three songs with the performers who...

You are asked to match three songs with the performers who made those songs famous. If you guess, the probability distribution for the number of correct matches is: $$\begin{array}{|lrrrr|}\hline \text { Probability } & .333 & .500 & 0 & .167 \\\text { Number correct } & 0 & 1 & 2 & 3 \\\hline\end{array}$$ What is the probability you get: a. Exactly one correct? b. At least one correct? C. Exactly two correct? d. Compute the mean, variance, and standard deviation of this distribution.

You are asked to match three songs with the performers who made those songs famous. If you guess, the probability distribution for the number of correct matches is:
$$\begin{array}{|lrrrr|}\hline \text { Probability } & .333 & .500 & 0 & .167 \\\text { Number correct } & 0 & 1 & 2 & 3 \\\hline\end{array}$$
What is the probability you get:
a. Exactly one correct?
b. At least one correct?
C. Exactly two correct?
d. Compute the mean, variance, and standard deviation of this distribution.

Key Concepts

Probability Distribution

A probability distribution assigns a probability to each outcome of a random variable. In this context, it tells us the likelihood that a certain number of matches (0, 1, 2, or 3) will be correct, ensuring the total probability sums to 1.

Complement Rule

The complement rule helps find the probability of an event by subtracting the probability of its complement from 1. For example, knowing the probability of getting no correct matches allows one to compute the probability of getting at least one correct match.

Exact vs. At Least Probabilities

Understanding the difference between 'exactly' and 'at least' is key. 'Exactly' refers to the probability of one specific number of successes, while 'at least' is the probability of that number or more, which can be determined by summing the probabilities for all relevant outcomes.

Expected Value

The expected value, or mean, is the weighted average of all possible values of the random variable, calculated by multiplying each outcome by its probability and summing these products. It gives a long?run average outcome over many trials.

Variance

Variance measures the spread of the random variable around the expected value. It is calculated by taking the average of the squared differences between each outcome and the mean, weighted by the probability of each outcome.

Standard Deviation

Standard deviation is the square root of the variance and provides a measure of dispersion in the same units as the original data. It indicates how much the outcomes typically deviate from the mean.

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