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What is Expected value?

The average value taken by a quantity determined by chance. It is found by multiplying each possible value by the probability of that value and adding the results. For example, the expected value of a die roll is (1 + 2 + … + 6) ÷ 6 = 3.5. The expected value of a sum is the sum of the expected values (linearity of expectation), and this holds even for quantities that are not independent. It is used for comparing countermeasures by expected loss, for decision making and for calculating averages in queueing.

Japanese name: 期待値

Expected value appears in 6 practice questions across 4 topics, including Probability, Statistics and Queueing, Business Strategy and Accounting Basics, Information Security Basics.

Field: Probability, Statistics and Queueing

Try it: questions about Expected value

There are 1,000 lottery tickets at ¥50 each: one wins a ¥10,000 prize, ten win ¥1,000 prizes, and the rest win nothing. Which of the following is the expected value of the prize per ticket?

  1. A−¥30
  2. B¥11
  3. C¥20
Show answer and explanation

Answer: C. ¥20

The expected value is the sum of each value multiplied by the probability of that value. The probability of ¥10,000 is 1/1,000 and that of ¥1,000 is 10/1,000, so 10,000 × 1/1,000 + 1,000 × 10/1,000 = 10 + 10 = ¥20. Subtracting the ticket price of ¥50 gives an expected gain or loss of −¥30 per ticket.

A company is considering investing in a new production line. A large investment (¥1 billion) is expected to bring revenue of ¥2.5 billion if demand is high (probability 0.6) and ¥800 million if demand is low (probability 0.4). A small investment (¥400 million) brings revenue of ¥700 million if demand is low; if demand is high, it brings ¥1.2 billion as is, but an additional investment of ¥500 million at that point raises revenue to ¥2.2 billion. If the policy that maximizes the expected profit (revenue minus investment) is adopted, which is that expected value?

  1. A¥820 million
  2. B¥900 million
  3. C¥1.2 billion
Show answer and explanation

Answer: B. ¥900 million

Working in units of ¥100 million, the expected profit of the large investment is 0.6 × 25 + 0.4 × 8 − 10 = 8.2, that is, ¥820 million. For the small investment, investing more when demand is high gives 22 − 5 = 17, which is more than the 12 without it, so the additional investment is chosen, and the expected value is 0.6 × 17 + 0.4 × 7 − 4 = 9.0, that is, ¥900 million. In a decision tree, the best choice is decided starting from the later decisions, and expected values are then compared working backward. Therefore the best policy is to start with the small investment and invest more if demand is high, with an expected value of ¥900 million.

Expected value in the explanations

Questions that mention Expected value (6)

  1. There are 1,000 lottery tickets at ¥50 each: one wins a ¥10,000 prize, ten win ¥1,000 prizes, and the rest win nothing. Which of the following is the expected value of the prize per ticket?
  2. A company is considering investing in a new production line. A large investment (¥1 billion) is expected to bring revenue of ¥2.5 billion if demand is high (probability 0.6) and ¥800 million if demand is low (probability 0.4). A small investment (¥400 million) brings revenue of ¥700 million if demand is low; if demand is high, it brings ¥1.2 billion as is, but an additional investment of ¥500 million at that point raises revenue to ¥2.2 billion. If the policy that maximizes the expected profit (revenue minus investment) is adopted, which is that expected value?
  3. Ten keys are stored in a hash table with 100 buckets. Each key is assigned independently to one of the 100 buckets with equal probability. Which of the following is the expected number of pairs of keys that are assigned to the same bucket?
  4. You arrive at a bus stop served every 10 minutes at a random time, without checking the timetable. The waiting time follows a uniform distribution from 0 to 10 minutes. Which of the following correctly combines the expected waiting time and the probability that the wait is 3 minutes or less?
  5. For an information system, it is estimated that an incident causing a loss of 20 million yen each time occurs 0.1 times per year. A countermeasure costing 0.5 million yen per year would reduce the frequency to 0.02 times per year. What is the net annual benefit of this countermeasure (the reduction in expected loss minus the cost of the countermeasure)?
  6. A three-point estimate of an activity's duration gives an optimistic value of 4 days, a most likely value of 7 days and a pessimistic value of 16 days. What is the expected duration given by the PERT formula?

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Related terms

Hash table Standard deviation Bayes' theorem Binomial distribution Conditional probability Confidence interval Correlation coefficient Exponential distribution Hypothesis testing Median

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