Textbook Question
Happiness In a survey sponsored by Coca-Cola, subjects were asked what contributes most to their happiness, and the table summarizes their responses. Does the table represent a probability distribution? Explain.
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5. Binomial Distribution & Discrete Random Variables
Binomial Distribution
2:04 minutesProblem 5.CRE.2b
Textbook Question
Kentucky Pick 4 In the Kentucky Pick 4 lottery game, you can pay \$1 for a “straight” bet in which you select four digits with repetition allowed. If you buy only one ticket and win, your prize is \$2500.
b. If you play this game once every day, find the mean number of wins in years with exactly 365 days.
Verified step by step guidance1
Step 1: Understand the problem. The Kentucky Pick 4 lottery involves selecting four digits with repetition allowed, meaning each digit can range from 0 to 9. The total number of possible combinations is calculated as 10^4, since there are 10 choices for each of the 4 digits.
Step 2: Calculate the probability of winning. Since there is only one winning combination out of the total possible combinations, the probability of winning on a single ticket is given by P(win) = 1 / (10^4).
Step 3: Determine the number of days played in a year. The problem states that the game is played once every day in a year with 365 days. Therefore, the total number of games played in a year is 365.
Step 4: Use the formula for the mean number of wins. The mean number of wins in a given number of trials is calculated as μ = n * P(win), where n is the number of trials (365 days) and P(win) is the probability of winning on a single ticket.
Step 5: Substitute the values into the formula. Replace n with 365 and P(win) with 1 / (10^4) to compute the mean number of wins. The result will represent the expected number of wins in a year with 365 days.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability
Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In the context of the Kentucky Pick 4 lottery, the probability of winning with a single ticket can be calculated based on the total number of possible combinations of four digits, considering that repetition is allowed.
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Introduction to Probability
Expected Value
Expected value is a statistical concept that provides a measure of the center of a probability distribution, representing the average outcome if an experiment is repeated many times. In this lottery scenario, the expected number of wins over a year can be calculated by multiplying the probability of winning by the number of plays (365 days).
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04:14Expected Value (Mean) of Random Variables
Mean
The mean, or average, is a fundamental statistical measure that summarizes a set of values by dividing the sum of those values by the number of values. In this case, the mean number of wins in a year can be derived from the expected value, indicating how many times a player can expect to win if they play the lottery daily.
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04:52Calculating the Mean
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