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Chapter 4: Problem 35

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 Two marbles are withdrawn randomly.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.

Short Answer

Expert verified
The expected value of the amount you win is $0.13, and the variance of the amount you win is $1.30^2.

Step by step solution

01

Calculate the probabilities of each outcome

To calculate the probabilities, we will use the number of ways each outcome can occur and the total number of possibilities. 1. Probability of drawing two red marbles (RR): \( P(RR) = \frac{5}{10} \times \frac{4}{9} = \frac{2}{9} \) 2. Probability of drawing two blue marbles (BB): \( P(BB) = \frac{5}{10} \times \frac{4}{9} = \frac{2}{9} \) 3. Probability of drawing one red and one blue marble (RB or BR): \( P(RB \text{ or } BR) = \frac{5}{10} \times \frac{5}{9} + \frac{5}{10} \times \frac{5}{9} = \frac{5}{9} \)
02

Calculate expected value

We will now calculate the expected value (E) of the amounts won, which is given by E = 危(P(outcome) 脳 amount_won). E = \( P(RR) \times \$1.10 + P(BB) \times \$1.10 - P(RB \text{ or } BR) \times \$1.00 \) Substitute the calculated probabilities: E = \( \frac{2}{9} \times \$1.10 + \frac{2}{9} \times \$1.10 - \frac{5}{9} \times \$1.00 = \$0.13 \) So, the expected value of the amount you win is $0.13.
03

Calculate the variance

The variance (V) is calculated using the formula V = 危(P(outcome) 脳 (amount_won - E)^2). V = \( P(RR) \times (\$1.10 - \$0.13)^2 + P(BB) \times (\$1.10 - \$0.13)^2 + P(RB \text{ or } BR) \times (-\$1.00 - \$0.13)^2 \) Substitute the calculated probabilities and expected value: V = \( \frac{2}{9} \times (\$0.97)^2 + \frac{2}{9} \times (\$0.97)^2 + \frac{5}{9} \times (-\$1.13)^2 = \$1.30^2 \) So, the variance of the amount you win is $1.30^2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Probability Theory
Probability theory is a branch of mathematics focused on analyzing random events. The essence of probability is to quantify the likelihood of various outcomes in processes that cannot be predicted with absolute certainty.

In the context of the exercise, you're dealing with a classic probability problem: determining the chances of drawing marbles from a box. Here, the total number of possible outcomes for drawing two marbles is 10 choose 2. Probability theory guides us through the process of calculating specific outcomes, like drawing two red marbles or a red and a blue marble. By assigning probabilities to these outcomes, we apply the fundamental rules of probability theory to predict what could happen in a random experiment.
Calculating Expected Value
Expected value is a concept used to determine the average outcome of a random event if it were to be repeated many times. In terms of the marble exercise, the expected value calculation tells you the average amount you can expect to win per draw over the long run.

The expected value is found by multiplying each possible outcome by the probability of that outcome, then summing all these products together. Mathematically, it's represented as E = 危(P(outcome) 脳 outcome value). For the marbles game, you multiply the winnings by the likelihood of drawing two marbles of the same color, then subtract the product of losing by the probability of drawing two different colors. This provides you with the expected value - the average win or loss per game.
Understanding Variance Calculation
Variance provides insight into the spread of a random variable's possible values around the expected value. It measures how much the results can vary from the expected average.

In practice, you calculate variance by finding the squared difference between the possible amounts won and the expected value, multiplying each by the probability of that outcome, then adding all these values together. In the marble drawing scenario, you're calculating the variance in winnings, which provides a sense of risk or volatility in the game. A higher variance means more unpredictability and a greater risk of divergence from what you expect to win on average. The formula is V = 危(P(outcome) 脳 (amount_won - E)^2). After substituting the probabilities and the expected value, you end up with the game鈥檚 variance.

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Most popular questions from this chapter

For a nonnegative integer-valued random variable \(N\), show that $$ E[N]=\sum_{i=1}^{\infty} P\\{N \geq i\\} $$

The probability of being dealt a full house in a hand of poker is approximately \(.0014\). Find an approximation for the probability that in 1000 hands of poker you will be dealt at least 2 full houses.

A coin that when flipped comes up heads with probability \(p\) is flipped until either heads or tails has occurred twice. Find the expected. number of flips.

In some military courts, 9 judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with probability \(.7\), whereas when the defendant is, in fact, innocent, this probability drops to \(.3\). (a) What is the probability that a guilty defendant is declared guilty when there are (i) 9 , (ii) 8 , and (iii) 7 judges? (b) Repeat part (a) for an innocent defendant. (c) If the prosecution attorney does not exercise the right to a peremptory challenge of a judge and if the defense is limited to at most two such challenges, how many challenges should the defense attorney make if he or she is 60 percent certain that the client is guilty?

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