Today's blog post is an introduction to the Expected value and hopefully would be helpful to the kids who would be attending the upcoming Expected Value Lecture.
The handouts/ books I referred to are Evan Chen's Probability handout , AOPS introduction to Counting and probability, Calt's Expected value handout, brilliant and this IIT Delhi handout.
Probability is Global
Expected Value:
- The expected value is the sum of the probability of each individual event multiplied by the number of times the event happens.
- It is denoted as
- We have
where is the value of the outcome and is the probability that occurs.
Problem 1: What is the expected value of the number that shows up when you roll a fair sided
dice?
Solution: Since it's a fair dice, we get each outcome to have equal probability i.e
Solution: Since it's a fair dice, we get each outcome to have equal probability i.e
So
Problem 2: Find the expected value of a roll on a fair sided dice, labelled from to
Solution: Since it's a fair dice, we get each outcome to have equal probability i.e So
Problem 3: Suppose you have a weighted coin in which heads comes up with probability
Solution:
Problem 4: At a raffle, tickets are sold at each for prizes of and You buy ticket. What is the expected value of your gain?
Solution:
Problem 5: Linda estimates the number of questions she answered correctly on a test. She answered correctly with probability correctly with probability and correctly with probability What is the expected value of the number of questions Linda answered correctly?
Solution:
Solution:
Problem 6: Mara is playing a game. There are two marbles in a bag. If she chooses the purple marble, she will win If she chooses the orange marble, she will win What is the expected value of Mara's winnings from the game?
Solution:
Solution:
Problem 7: In the casino game roulette, a wheel with spaces ( red, black, and green) is spun. In one possible bet, the player bets on a single number. If that number is spun on the wheel, then they receive (their original ). Otherwise, they lose their On average, how much money should a player expect to win or lose if they play this game repeatedly?
Solution:
Solution:
Problem 8: In a certain state's lottery, balls numbered through are placed in a machine and six of them are drawn at random. If the six numbers are drawn match the numbers that a player had chosen, the player wins If they match numbers, then win It costs to buy a ticket. Find the expected value.
Solution:
Solution:
Linearity of Expectation:
If there exist variables independent or dependent,
Also holds when independent.
Problem 9: What is the expected value of the sum of two dice rolls?
Solution: Let the expected value of the first dice be and the second dice be
So
Problem 10: Caroline is going to flip fair coins one after the other. If she flips heads, she will be paid . What is the expected value of her payout?
Solution: Let be if heads and Also, denote as the outcome of the th coin flip.
Solution: Let
So
Problem 11: Sammy is lost and starts to wander aimlessly. Each minute, he walks one meter forward with probability , stays where he is with probability , and walks one meter backward with probability . After one hour, what is the expected value for the forward distance (in meters) that Sammy has travelled?
Solution: Let be the move sammy does in minute. Note that
Solution: Let
So
Problem 12: independent, fair coins are tossed in a row. What is the expected number of consecutive HH pairs?
Solution: So consider the consecutive pairs. Let denote the th coin in the row.. Then we consider the pairs
Solution: So consider the consecutive pairs. Let
Now, let
Note that
Note that
Hence, even though they are dependent, by linearity of expectation,
Problem 13: Suppose that and each randomly, and independently,
choose of objects. Find the expected number of objects chosen by both and
Solution: Let be the number of objects chosen by both A and B. Then let
So Alternatively, we have
So
So
Problem 14: At a nursery, babies sit in a circle. Suddenly, each baby randomly pokes either the baby to its left or to its right. What is the expected value of the number of unpoked babies?
Solution: Let the babies be
Note that any pair ( defining when unpoked)
And then we do linearity of Expectation.
It's the famous paradox game. :P
Problem 15: You are playing a game in which prize pool starts at On every turn, you flip a fair coin. If you flip head, then the prize pool doubles. If tails, the game ends.
Solution: Note that
Problem 16: Two random, not necessarily distinct, permutations of the digits are selected and added together. What is the expected value of this sum?
Solution: Thanks to Pranav for the write up.
Let the permutations be And the sums be
Total number of permutations of . Total number of distinct sums .
Let be a random variable representing sum of two permutations of taken at random. Then, . Now, we have to calculate . Clearly,
The following proof is from the calt handout! The handout is very nice!!!!
Theorem: If the probability of a variable occurring is then the expected number of times we must repeat the event so that we get is .
Proof: Let be the number of times we would have to repeat to get
So
multiplying by and subtracting,
Yeah, that is it! Hope you enjoyed reading this post!
Sunaina 💚
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