Expected Value in North Carolina’s Pick 4 Game In North Carolina’...

4. Probability

Fundamental Counting Principle

Statistics

4. Probability

Fundamental Counting Principle

1:27 minutes

Problem 5.1.30a

Textbook Question

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?

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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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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?