Expected value 𝑬(𝒙) - (Mean of a random variable)
Expected Value – The mean of a random variable. It is often used to predict how much is
likely to be gained or lost based on making a certain decision or taking a specific action. If the
expected value is positive, a gain is expected. If it is negative, a loss is expected. The expected
value represents the average gain (profit) or loss if the same decision is repeated over and
over again. In application, the expected value is typically illustrated through ‘games of
chance’. In general, it can be calculated as 𝐸(𝑥) = ∑ [𝑥 ∙ 𝑃(𝑥)], or
𝐸(𝑥) = ∑(𝑃𝑟𝑜𝑓𝑖𝑡)(𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑃𝑟𝑜𝑓𝑖𝑡) − ∑(𝐶𝑜𝑠𝑡)(𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐶𝑜𝑠𝑡)
Examples -
Life Insurance Policies –
An insurance company sells a one-year term life insurance policy to a 75-year old woman. The woman
pays a premium of $750. If she dies within 1 year, the insurance company will pay $25,000 to her beneficiary.
According to the 2014 Social Security Actuarial Life Table, the probability that a 75-year old woman will be
alive in 1 year is 0.9746. Find the expected value of the insurance company’s profit on the policy. Interpret the
result.
𝐸(𝑥) 𝑜𝑓 𝑃𝑟𝑜𝑓𝑖𝑡 = ($750)(0.9746) − ($25,000 − $750)(1 − 0.9746) = $115
The amount The probability The amount The
required to that the the insurance probability of
purchase the woman lives company the woman
policy for at least 1 “loses” if the dying within 1
year woman dies year
Interpretation - This indicates that if the insurance company sells many policies like this one, the
company can expect on average to earn a $115 profit per policy. (Gain)
Lottery Tickets –
In a state lottery, you pay $1 and pick a number from 000 to 999. If your number comes up, you win
$600. The probability of winning is 0.001. What is the expected value of your profit? Is this a gain or loss?
𝐸(𝑥) = ($600 − 1)(0.001) − ($1)(0.999) = − $0.40 (Loss)
The amount The The cost of The probability
that you can probability a ticket of losing
profit of winning
Practice Problems -
Raffle Tickets –
You purchase a $5 raffle ticket for a fundraiser. A total of 500 tickets are sold. 5 winners will be
drawn. 1st prize will be $250. 2nd prize will be $150. 3rd prize will be $100. The 4th and 5th place
winners will each receive $50. What is the expected value of your winnings?
Answer: E(x) = ($245) + ($145) + ($95) + ($45) + ($45) − ($5)
= - $3.80 (Loss)
SAT Scores –
a. You are taking an SAT test. If you get the question right, you gain one point, but if you get the question
wrong, you lose ¼ of a point. You have no idea what the correct answer is and randomly take a guess
from the 5 choices given. What is the expected value for the number of points you get?
Answer: E(x) = (1)(0.2) – (1/4)(0.8) = 0
b. On the next SAT test question, you confidently eliminate 2 of the choices and randomly guess from the
3 remaining options. What is the expected value for the number of points you get?
Answer: E(x) = (1)(1/3) – (1/4)(2/3) = 0.1666…. (Gain)
Gambling –
In the game of “Craps”, a pair of dice are rolled. You can bet $1 that the sum of the dice will be 2
(“snake eyes”). The probability of winning is 1/36. If you win, the profit is $30. What is the expected
value of your profit?
Answer: E(x) = ($30)(1/36) – ($1)(35/36) = - $0.14 (Loss)
Business Opportunity –
You are considering investing $10,000 into a start-up company. You estimate that you have a 0.25
probability of a $20,000 loss, a 0.20 probability of a $10,000 profit, a 0.15 probability of a $50,000
profit, and a 0.40 probability of breaking even (no profit). What is the expected value of the profit?
Should you follow through on this investment?
Answer: E(x) = ($10,000)(0.20) + ($50,000)(0.15) +($0)(0.40) – ($20,000)(0.25)
= $4500 (Yes)
