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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
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Tutti i libri di testoTriola 14th EditionCh. 5 - Discrete Probability DistributionsProblema 5.1.30a
Capitolo 5, Problema 5.1.30a

Expected Value in North Carolina’s Pick 4 Game In North Carolina’s Pick 4 lottery game, you can pay \$1 to select a four-digit number from 0000 through 9999. If you select the same sequence of four digits that are drawn, you win and collect \$5000.


a. How many different selections are possible?

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Step 1: Understand the problem. The goal is to calculate the total number of possible four-digit combinations in the North Carolina Pick 4 lottery game. Each digit can range from 0 to 9, and the sequence matters.
Step 2: Recognize that each digit in the four-digit number can independently take on 10 possible values (0 through 9). This is a fundamental principle of counting in probability.
Step 3: Use the multiplication rule for independent events. Since there are 4 digits and each digit has 10 possible values, the total number of combinations is calculated as \( 10 \times 10 \times 10 \times 10 \).
Step 4: Express the calculation in exponential form for simplicity. The total number of combinations can also be written as \( 10^4 \), which represents 10 raised to the power of 4.
Step 5: Conclude that the result of \( 10^4 \) gives the total number of different selections possible in the game. This is the number of unique four-digit sequences that can be chosen.

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Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. In the context of the Pick 4 lottery game, it helps determine the total number of possible selections of four-digit numbers. Since each digit can range from 0 to 9, combinatorial principles allow us to calculate the total combinations available.

Probability

Probability is the measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In the Pick 4 game, understanding probability is essential for evaluating the chances of winning based on the total number of possible outcomes. This concept helps players assess the risk versus reward of participating in the lottery.
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Introduction to Probability

Random Variables

A random variable is a numerical outcome of a random phenomenon, often used in statistics to quantify outcomes. In the context of the Pick 4 game, the winning number can be considered a random variable, as it is drawn randomly from a set of possible outcomes. Understanding random variables is crucial for analyzing the expected value and potential winnings in games of chance.
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Intro to Random Variables & Probability Distributions
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One of Mendel’s famous experiments with peas resulted in 580 offspring, and 152 of them were yellow peas. Mendel claimed that under the same conditions, 25% of offspring peas would be yellow. Assume that Mendel’s claim of 25% is true, and assume that a sample consists of 580 offspring peas.


a. Use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high. Based on the results, is the result of 152 yellow peas either significantly low or significantly high?


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Using Probabilities for Significant Events


a. Find the probability of getting exactly 1 match.

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In Exercises 31 and 32, assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods (as in one of Mendel’s famous experiments).


Hybrids Assume that offspring peas are randomly selected in groups of 16.


a. Find the mean and standard deviation for the numbers of peas with green pods in the groups of 16.

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Politics The County Clerk in Essex, New Jersey, was accused of cheating by not using randomness in assigning the order in which candidates’ names appeared on voting ballots. Among 41 different ballots, Democrats were assigned the desirable first line 40 times. Assume that Democrats and Republicans are assigned the first line using a method of random selection so that they are equally likely to get that first line.


a. Use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high. Based on the results, is the result of 40 first lines for Democrats significantly high?

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In Exercises 25–28, find the probabilities and answer the questions.


Internet Voting Based on a Consumer Reports survey, 39% of likely voters would be willing to vote by Internet instead of the in-person traditional method of voting. For each of the following, assume that 15 likely voters are randomly selected.


a. What is the probability that exactly 12 of those selected would do Internet voting?

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In Exercises 29 and 30, assume that different groups of couples use the XSORT method of gender selection and each couple gives birth to one baby. The XSORT method is designed to increase the likelihood that a baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5.


Gender Selection Assume that the groups consist of 36 couples.


a. Find the mean and standard deviation for the numbers of girls in groups of 36 births.


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