A box contains 5 red and 5 blue marbles. Two marbles are...

A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win $\$ 1.10 ;$ if they are different colors, then you win $-\$ 1.00 .$ (That is, you lose $\$ 1.00 .$ ) Calculate (a) the expected value of the amount you win; (b) the variance of the amount you win.

A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win $\$ 1.10 ;$ if they are different colors, then you win $-\$ 1.00 .$ (That is, you lose $\$ 1.00 .$ ) Calculate
(a) the expected value of the amount you win;
(b) the variance of the amount you win.

Key Concepts

Probability Calculation Using Combinatorics

Probability calculation in the context of drawing objects (such as marbles) without replacement often employs combinatorial methods. By using combinations, one can determine the number of favorable outcomes and total possible outcomes, which allows for the accurate computation of probabilities for events occurring under a finite sample space.

Random Variables and Their Distributions

A random variable is a variable that assigns numerical values to the outcomes of a random experiment. In this context, the winnings are represented by a random variable that takes on specific monetary values based on the outcome of the draw. The distribution of this random variable is constructed by associating each outcome with a specific probability, which is fundamental to calculating expected value and variance.

Expected Value

Expected value is a statistical measure that represents the average outcome of a random experiment when it is repeated many times. It is calculated by multiplying each possible outcome by its corresponding probability and then summing all those products, thereby providing a single number which encapsulates the central tendency of the distribution of outcomes.

Variance

Variance is a measure of the dispersion or spread of a set of outcomes around the expected value. It is calculated by taking the average of the squared differences between each outcome and the expected value, which quantifies the degree of variability or uncertainty in the outcomes of the random experiment.

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