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Chapter 14: Problem 29

You play two games against the same opponent. The probability you win the first game is 0.4 . If you win the first game, the probability you also win the second is 0.2 . If you lose the first game, the probability that you win the second is 0.3 . a. Are the two games independent? Explain. b. What's the probability you lose both games? c. What's the probability you win both games? d. Let random variable \(X\) be the number of games you win. Find the probability model for \(X\). e. What are the expected value and standard deviation?

Short Answer

Expert verified
a) The games are not independent. b) The probability of losing both games is 0.42. c) The probability of winning both games is 0.08. d) The probability model for the number of games won is \(P(X = 0) = 0.42, P(X = 1) = 0.44, P(X = 2) = 0.08\). e) The expected value of games won is 0.52 and standard deviation is 0.6.

Step by step solution

01

Independence of the games

Two events are considered independent if the occurrence of one event does not affect the probability of the other event. Here, the probability of winning the second game changes based on the result of the first game. Therefore, the games are not independent.
02

Probability of losing both games

The probability of losing the first game is \(1 - 0.4 = 0.6\). Given that the first game is lost, the probability of losing the second game is \(1 - 0.3 = 0.7\). Using the rule of multiplication for independent events, the probability of losing both games is \(0.6 * 0.7 = 0.42\).
03

Probability of winning both games

The probability of winning the first game is \(0.4\). Given that the first game is won, the probability of winning the second game is \(0.2\). Using the rule of multiplication for independent events, the probability of winning both games is \(0.4 * 0.2 = 0.08\).
04

Probability model for X

The random variable \(X\) can be the number of games won, so \(X\) can be \(0\), \(1\), or \(2\). The probability that \(X = 0\) is the probability of losing both games, which is \(0.42\). The probability that \(X = 1\) is the sum of the probabilities of winning the first game and losing the second, and losing the first game and winning the second. This adds up to \(0.4 * (1 - 0.2) + 0.6 * 0.3 = 0.44\). The probability that \(X = 2\), the probability of winning both games is \(0.08\). So, the probability model for a number of games won is \(P(X = 0) = 0.42, P(X = 1) = 0.44, P(X = 2) = 0.08\).
05

Expected value and standard deviation

The expected value, \(E(X)\), is calculated as \(\Sigma [x * P(x)]\), and standard deviation, \(\sigma\), is calculated as \(\sqrt{\Sigma [(x - E(X))^2 * P(x)]}\). For \(E(X)\), we'd calculate as \(0*0.42 + 1*0.44 + 2*0.08 = 0.52\). For \(\sigma\), we'd calculate as \(\sqrt{(0 - 0.52)^2 * 0.42 + (1 - 0.52)^2 * 0.44 + (2 - 0.52)^2 * 0.08} = 0.6\)

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Key Concepts

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

Independent Events
Understanding independent events is crucial in the study of probability. These are events where the occurrence of one does not influence the likelihood of the other. For example, think about flipping a coin and rolling a die. Whether the coin lands on heads or tails does not affect the number on the die when it lands. However, the example from the exercise shows a different situation. Here, the probability of winning the second game changes depending on the outcome of the first game. This interdependence tells us that the outcomes of these games are not independent events. When events are not independent, you can't simply multiply their probabilities to find the likelihood of both occurring together, which is a common mistake for students starting with probability.
Probability Model
A probability model is a mathematical representation of a random phenomenon. It includes all possible outcomes of the event and the probabilities associated with each outcome. For our game example, we create a model that helps us understand the different possibilities of winning and losing the games. Here, the random variable \(X\), which represents the number of games won, can take values 0, 1, or 2. We calculate the probabilities for each of these outcomes by considering the given conditions. By crafting a probability model, you create a useful tool that not only guides you in predicting outcomes but also is foundational when calculating expected values and standard deviations, which bring more depth to understanding the nature of your random events.
Expected Value and Standard Deviation
Moving onto expected value and standard deviation, these statistics give you insight into the behavior of a random variable. The expected value, or mean, represents what you would predict to happen on average if the random scenario was repeated many times. It's essential for projecting probable outcomes in various fields like finance, insurance, and more. In our game scenario, the expected value \(E(X)\) of the random variable \(X\) is 0.52, suggesting that, on average, you will win slightly more than half a game per play in the long run.

The standard deviation, on the other hand, measures the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In statistics, the standard deviation is a key measure because it quantifies uncertainty. In our example, the standard deviation is 0.6, which indicates that the number of games won per play will typically fluctuate within a range around the average wins, illustrating the randomness of the game's outcome.

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Most popular questions from this chapter

The bicycle shop in Exercise 50 will be offering 2 specially priced children's models at a sidewalk sale. The basic model will sell for $$\$ 120$$ and the deluxe model for $$\$ 150 .$$ Past experience indicates that sales of the basic model will have a mean of 5.4 bikes with a standard deviation of 1.2 , and sales of the deluxe model will have a mean of 3.2 bikes with a standard deviation of 0.8 bikes. The cost of setting up for the sidewalk sale is $$\$ 200$$. a. Define random variables and use them to express the bicycle shop's net income. b. What's the mean of the net income? c. What's the standard deviation of the net income? d. Do you need to make any assumptions in calculating the mean? How about the standard deviation?

At a certain coffee shop, all the customers buy a cup of coffee; some also buy a doughnut. The shop owner believes that the number of cups he sells each day is Normally distributed with a mean of 320 cups and a standard deviation of 20 cups. He also believes that the number of doughnuts he sells each day is independent of the coffee sales and is Normally distributed with a mean of 150 doughnuts and a standard deviation of 12 . a. The shop is open every day but Sunday. Assuming day-to-day sales are independent, what's the probability he'll sell more than 2000 cups of coffee in a week? b. If he makes a profit of 50 cents on each cup of coffee and 40 cents on each doughnut, can he reasonably expect to have a day's profit of over $$\$ 300 ?$$ Explain. c. What's the probability that on any given day he'll sell a doughnut to more than half of his coffee customers?

A farmer has \(100 \mathrm{lb}\) of apples and \(50 \mathrm{lb}\) of potatoes for sale. The market price for apples (per pound) each day is a random variable with a mean of 0.5 dollars and a standard deviation of 0.2 dollars. Similarly, for a pound of potatoes, the mean price is 0.3 dollars and the standard deviation is 0.1 dollars. It also costs him 2 dollars to bring all the apples and potatoes to the market. The market is busy with eager shoppers, so we can assume that he'll be able to sell all of each type of produce at that day's price. a. Define your random variables, and use them to express the farmer's net income. b. Find the mean. c. Find the standard deviation of the net income. d. Do you need to make any assumptions in calculating the mean? How about the standard deviation?

You bet! You roll a die. If it comes up a \(6,\) you win $$\$ 100$$. If not, you get to roll again. If you get a 6 the second time, you win $$\$ 50$$. If not, you lose. a. Create a probability model for the amount you win. b. Find the expected amount you'll win. c. What would you be willing to pay to play this game?

Pick a card, any card You draw a card from a deck. If you get a red card, you win nothing. If you get a spade, you win $$\$ 5$$. For any club, you win \(\$ 10\) plus an extra \(\$ 20\) for the ace of clubs. a. Create a probability model for the amount you win. b. Find the expected amount you'll win. c. What would you be willing to pay to play this game?
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