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Chapter 13: Problem 28

Playing the lottery. New York State's "Quick Draw" lottery moves right along. Players choose between 1 and 10 numbers from the range 1 to \(80 ; 20\) winning numbers are displayed on a screen every four minutes. If you choose just one number, your probability of winning is \(20 / 80\), or \(0.25\). Lester plays one number eight times as he sits in a bar. What is the probability that all eight bets lose?

Short Answer

Expert verified
The probability that all eight bets lose is approximately 0.10.

Step by step solution

01

Determine the probability of losing one play

Lester's probability of winning with one number is given as \(0.25\). Therefore, the probability \(P(L)\) of losing one play is \(1 - 0.25 = 0.75\).
02

Calculate the probability of losing eight consecutive plays

Since each play is independent, the probability of losing eight times in a row is the probability of losing one time raised to the eighth power. Thus, the probability \(P(8L)\) is \(0.75^8\).
03

Compute the final probability

Calculate \(0.75^8\) using a calculator. \(0.75^8 = 0.1001129144\). Round this number to two decimal places to get the probability of all eight bets losing.

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Key Concepts

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

Lottery
Lotteries are a popular game of chance involving the selection of numbers. In the New York State's "Quick Draw" lottery, players can choose between 1 and 10 numbers from a pool ranging from 1 to 80. Every few minutes, 20 winning numbers are randomly displayed on a screen. This high-paced lottery offers players a quick shot at winning. Players' odds and strategies vary depending on the number of picks they make. Choosing fewer numbers, such as just one, lowers the odds but can make calculations around probability easier to understand.
Independent Events
In probability, independent events are those whose outcomes do not affect each other. This means that no matter how many times you play the lottery, the outcome of one draw doesn't change the odds for the next. For Lester, each time he picks his number in the "Quick Draw" lottery, his chances remain constant since each play is independent. Whether he wins or loses one round does not affect the probability of winning or losing in the subsequent rounds. Understanding this is key to calculating probabilities over multiple plays.
Probability Calculation
Calculating the probability involves understanding the likelihood of an event occurring. In the "Quick Draw" lottery, the probability of Lester winning with one number is 0.25. To find the probability of losing just once, you take the complement of winning, which is 1 minus the probability of winning: thus, 1 - 0.25 = 0.75.

When calculating the probability of losing multiple times, since each play is independent, you multiply the probability of losing one play by itself, raised to the power of the number of plays. Therefore, to find the probability of Lester losing all eight of his bets, you calculate 0.75 raised to the power of 8, which simplifies to 0.75^8.
Statistical Analysis
Statistical analysis in the context of the lottery involves computing probabilities and understanding odds to make informed decisions. It helps to quantify the chances of different outcomes. Knowing that the probability of Lester losing all eight plays is approximately 0.10 (after rounding) allows him to assess his risks.

This analysis reinforces the understanding of how independent events work together and highlights the improbability of a string of losses or wins over several rounds. Using statistical tools helps players to not only play smart but also manage their expectations reasonably during gambling activities.

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Most popular questions from this chapter

White Cats and Deafness. Although cats generally possess an acute sense of hearing, due to an anomaly in their genetic makeup, deafness among white cats with blue eyes is quite common. Approximately \(95 \%\) of the general cat population are non-white cats (i.e., not pure white), and congenital deafness is extremely rare in non-white cats. However, among white cats, approximately \(75 \%\) with two blue eyes are deaf, \(40 \%\) with one blue eye are deaf, and only \(19 \%\) with eyes of other colors are deaf. Additionally, among white cats, approximately \(23 \%\) have two blue eyes, \(4 \%\) have one blue eye, and the remainder have eyes of other colors. 9 (a) Draw a tree diagram for selecting a white cat (outcomes: one blue eye, two blue eyes, or eyes of other colors) and deafness (outcomes: deaf or not deaf). (b) What is the probability that a randomly chosen white cat is deaf?

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