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Chapter 9: Problem 17

Suppose that in a certain scratch-ticket lottery game, the probability of winning with the purchase of one card is 1 in 500 , or \(0.2 \%\); hence, the probability of losing is \(100 \%-0.2 \%=99.8 \%\). But what if you buy more than one ticket? One way to calculate the probability that you will win at least once if you buy \(n\) tickets is to subtract from \(100 \%\) the probability that you will lose on all \(n\) cards. This is an easy calculation; the probability that you will lose two times in a row is \((99.8 \%)(99.8 \%)=\) (a) What is the probability that you will win at least once if you play three times? (b) Find a formula for \(P(n)\), the percentage chance of winning at least once if you play the game \(n\) times. (c) How many tickets must you buy in order to have a \(25 \%\) chance of winning? A \(50 \%\) chance? (d) Does doubling the number of tickets you buy also double your chances of winning? (e) Sketch a graph of \(P(n) .\) Use \([0,100]\) as the range of the graph. Explain the practical significance of any asymptotes.

Short Answer

Expert verified
For (a) the probability of winning at least once in 3 games is \(1 - (0.998)^3\).\nFor (b), the probability of winning at least once in \(n\) games is \(1 - (0.998)^n\). \nFor (c), one would solve \((0.998)^n = 0.75\) and \((0.998)^n = 0.50\) to find how many tickets they need to buy to have a 25% and 50% win chance respectively. \nFor (d), doubling the number of games does not double the chance of winning due to the use of exponent in the formula. \nFor (e), as \(n\) gets larger, \(P(n)\) approaches 1, giving a horizontal asymptote at y = 1.

Step by step solution

01

Calculating individual probabilities

To begin with, understand that in this game, the probability of winning once is \(0.2\%\) and the probability of losing is \(99.8\%\). We first need to convert these percentages into decimals because it is easier to work with. We get \(0.002\) and \(0.998\) respectively.
02

Probability of winning once in 3 tries

For part (a) we need to find the probability that we will win at least once if we play 3 times. The easiest way to do this is to subtract from 1 the probability that we will lose 3 times in a row. Hence, the probability of winning at least once in 3 tries can be given as \(1 - (0.998)^3\).
03

Formula for probability

For part (b), we find a general formula that applies for \(n\) games. Following similar logic just like in step 2, the probability of winning at least once in \(n\) games is given by \(1 - (0.998)^n\).
04

Calculating number of tickets for a certain win

For part (c), we need to calculate how many tickets we need to buy to have a \(0.25\) and \(0.50\) win chance. To do so, we set the formula from step 3 equal to these probabilities and solve for \(n\). For a \(0.25\) chance of winning, \((0.998)^n = 0.75\). Similarly, for a \(0.50\) chance of winning, \((0.998)^n = 0.50\). To solve this, you can use a logarithm and get \(n\).
05

Effect of doubling the number of tickets

For part (d), you may note that doubling the number of games does not double the chance of winning. This is due to the use of exponent in the formula.
06

Sketching the graph

For (e), we need to sketch the function \(P(n) = 1 - (0.998)^n\). Note that as \(n\) gets larger, \(P(n)\) approaches 1, but never exceeds it. Therefore, the function has a horizontal asymptote at \(P(n) = 1\).

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Key Concepts

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

Winning Probability
The concept of winning probability is crucial when considering chances in games like lotteries or scratch cards. In simple terms, winning probability refers to how likely it is for you to win in a particular game. To better understand this, let's take an example of a scratch-ticket lottery game.
Suppose you purchase one ticket. If the chance of winning is 0.2%, this is rather slim. It means that for every 500 tickets, you might win once. However, what happens when you purchase more tickets?
To find out the probability of winning at least once when buying multiple tickets, one must subtract the probability of losing on all tickets from 100%. For example, if you buy 3 tickets, your collective chance of winning at least once increases. This increase is calculated as the complement of losing all three times: \(1 - (0.998)^3\).
In essence, the more tickets you buy, the higher your probability of winning at least once. However, it's crucial to note this doesn't mean you will win; just that your chances increase statistically.
Lottery Probability
Lottery probability involves understanding how likely you are to succeed in random drawing games. Typically, these games have very low probabilities of winning because they rely heavily on chance. In our scratch-ticket example, the probability of losing with one ticket purchase is quite high, at 99.8%.
However, probabilities can be quite deceptive. If you wish to enhance your chance of winning, you might think buying more tickets is a surefire way. While this increases your winning probability, it's not linear. Doubling your ticket purchases doesn't double your chances of winning because probabilities compound exponentially.
  • Probability of losing continually diminishes as more tickets are bought, making winning more plausible.
  • For mathematicians, these considerations form the basis of creating formulas, such as \(1 - (0.998)^n\), to calculate probabilities over several tickets.
Understanding these concepts helps demystify the odds associated with lottery games.
Exponential Decay in Probability
Exponential decay in probability refers to how rapidly the chance of losing diminishes as more attempts are made. This principle can be observed in games of chance where continuous trials impact the probability of an outcome occurring.
In our lottery case, each additional ticket purchased represents a new trial. The probability of losing n times, denoted as \((0.998)^n\), highlights this decay.
  • With every ticket added, the term \((0.998)^n\) becomes significantly smaller, reducing the probability of losing.
  • Thus, the probability of winning at least once increases exponentially, approaching but never reaching certainty (a probability of 1).
This concept is represented graphically by an asymptote at 1, showing that while your chance of winning grows with more tickets, it never guarantees a win. Understanding this decay informs strategic decisions in games and helps set realistic expectations.

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Most popular questions from this chapter

The number of bacteria in a certain culture is known to triple every day. Suppose that at noon today there are 200 bacteria. (a) Construct a table of values to find a function that gives the number of bacteria after \(t\) days. (b) Approximately what was the population count at noon yesterday? At noon 4 days ago? (c) From now on, suppose the population at noon today is called \(B_{0}\) rather than being specifically 200 . Find a function that gives the number of bacteria after \(t\) days. (d) Express the number of bacteria as a function of \(w\), where \(w\) is time measured in weeks. (e) How many bacteria will be present at noon one week from today?

Factor \(b^{x}\) out of the following expressions. Check your answer by multiplying out. $$ b^{3 x}-(2 b)^{-1+2 x} $$

In the 1980 s the small town of Old Bethpage, New York, made the front page of the New York Times magazine section as an illustration of what was termed a "dying suburb." In Old Bethpage schools are being converted to nursing homes as the population ages and the baby boomers move out. Suppose that the number of school-age children in 1980 was \(C_{0}\), and was decreasing at a rate of \(6 \%\) per year. Let's assume that the number of school-age children continues to drop at a rate of \(6 \%\) each year. Let \(C(t)\) be the number of school-age children in Old Bethpage \(t\) years after 1980 . (a) Find \(C(t)\). (b) Express the number of school-age children in Old Bethpage in 1994 as a percentage of the 1980 population. (c) Use your calculator to estimate the year in which the population of school-age children in Old Bethpage will be half of its size in 1980 .

According to a report from the General Accounting Office, during the 14 -year period between the school year \(1980-1981\) and the school year \(1994-1995\), the average tuition at four-year public colleges increased by \(234 \%\). During the same period, average household income increased by \(82 \%\), and the Labor Department's Consumer Price Index (CPI) increased by 74\%. (Boston Globe, August 16, 1996.) (a) Assuming exponential growth, determine the annual percentage increase for each of these three measures. (b) The average cost of tuition in \(1994-1995\) was \(\$ 2865\) for in-state students. What was it in \(1980-1981\) ? (c) Starting with an initial value of one unit for each of the three quantities, average tuition at four-year public colleges, average household income, and the Consumer Price Index, sketch on a single set of axes the graphs of the three functions over this 14 -year period. (d) Suppose that a family has two children born 14 years apart. In \(1980-1981\), the tuition cost of sending the elder child to college represented \(15 \%\) of the family's total income. Assuming that their income increased at the same pace as the average household, what percent of their income was needed to send the younger child to college in \(1994-1995 ?\)

Let \(D(t), H(t)\), and \(J(t)\) represent the annual salaries (in dollars) of David, Henry, and Jennifer, and suppose that these functions are given by the following formulas, where \(t\) is in years. \(t=0\) corresponds to this year's salary, \(t=1\) to the salary one year from now, and so on. The domain of each function is \(t=0,1,2, \ldots\) up to retirement. $$ \begin{array}{c} D(t)=40,000+2500 t \\ H(t)=50,000(0.97)^{t} \\ J(t)=40,000(1.05)^{t} \end{array} $$ (a) Describe in words how each employee's salary is changing. (b) Suppose you are just four years away from retirement-you'll collect a salary for four years, including the present year. Which person's situation would you prefer to be your own? (c) If you are in your early twenties and looking forward to a long future with the company, which would you prefer?
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