Law of Large Numbers in Examples

Example 1: A fair six-sided die is rolled repeatedly. What is the expected average value of the outcomes as the number of rolls increases

Solution:

Expected value of rolling a fair six-sided die is the average of the numbers 1 through 6, which is (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5

According to the Law of Large Numbers, as the number of rolls increases, the average outcome will approach 3.5

Example 2: In a game, a player flips a fair coin. If it lands heads, the player wins $1; if it lands tails, the player loses $1. What is the expected average profit/loss for the player as the number of flips increases?

Solution:

Expected value of flipping a fair coin is 0.5 for heads and 0.5 for tails

Expected profit/loss for each flip is (0.5 × $1) + (0.5 × -$1) = $0

Therefore, the expected average profit/loss for the player as the number of flips increases approaches $0 by the Law of Large Numbers

Example 3: A bag contains 20 red balls and 30 blue balls. A ball is drawn from the bag, and the color is noted. The ball is then returned to the bag, and the process is repeated. What is the expected proportion of red balls drawn as the number of draws increases?

Solution:

Probability of drawing a red ball on any single draw is 20/(20+30) = 20/50 = 2/5

By the Law of Large Numbers, as the number of draws increases, the proportion of red balls drawn will approach 2/5

Example 4: A factory produces light bulbs, and historical data show that 5% of the bulbs are defective. If a random sample of bulbs is taken from the production line, what is the expected proportion of defective bulbs as the sample size increases?

Solution:

Probability of selecting a defective bulb from the production line is 0.05

As the sample size increases, by the Law of Large Numbers, the proportion of defective bulbs in the sample will approach 0.05

Example 5: In a game, a player rolls two fair six-sided dice and wins $10 if the sum of the dice is 7, and loses $5 otherwise. What is the expected average profit/loss for the player as the number of rolls increases?

Solution:

There are 6 possible outcomes when rolling two fair six-sided dice, and only one of these outcomes results in a sum of 7

So, the probability of winning $10 is 1/6, and the probability of losing $5 is 5/6. The expected profit/loss per roll is (1/6 × $10) + (5/6 × -$5) = -$5/6

Therefore, the expected average profit/loss for the player as the number of rolls increases approaches -$5/6 by the Law of Large Numbers

Example 6: A student takes multiple-choice quizzes with 5 questions, each with 4 answer choices. If the student randomly guesses the answers to all questions, what is the expected average score as the number of quizzes increases?

Solution:

Each question has 4 answer choices, so the probability of guessing the correct answer to any question is 1/4

Since there are 5 questions, the expected score for each quiz is (1/4 × 5) = 5/4

Therefore, the expected average score for the student as the number of quizzes increases approaches 5/4 by the Law of Large Numbers

Law of Large Numbers (LLN) is a mathematical theorem that states the average of the results obtained from many independent random samples.

In this article, we have discussed the Law of Large Numbers definition, its limitations, examples and others in detail.