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Expected value

Science and society, grade 11 (3 units) Lesson 14 of 16Learning

Expected value

The mean gain in a game

To know whether a game pays, work out how much it gains on average per turn – if it is played many times.

A fair has a wheel of fortune with 10 equal sectors:
one sector’s prize is 50 ₪, three sectors’ prize is 10 ₪, and the other six sectors win nothing.
A turn costs 10 ₪.

The gain is the prize minus the price of the turn:

Gain (₪)400−10\displaystyle -10
Probability0.1\displaystyle 0.10.3\displaystyle 0.30.6\displaystyle 0.6

In 1,000 turns there are about 100 gains of 40 ₪, 300 gains of 0, and 600 losses of 10 ₪:

100⋅40+300⋅0+600⋅(−10)1000=4000−60001000=−2\frac{100\cdot40+300\cdot0+600\cdot(-10)}{1000}=\frac{4000-6000}{1000}=-2

On average, the player loses 2 ₪ every turn.

The formula of the expected value

The expected value is the mean of the results of an experiment repeated many times. Multiply each value by its probability and add – as in a weighted mean whose weights are the probabilities.

E=x1p1+x2p2+⋯+xkpk\displaystyle E=x_1p_1+x_2p_2+\dots+x_kp_k

For the wheel of fortune:

E=40⋅0.1+0⋅0.3+(−10)⋅0.6=4+0−6=−2E=40\cdot0.1+0\cdot0.3+(-10)\cdot0.6=4+0-6=-2
  • A negative expected value – on average you lose; a positive one – on average you gain.
  • In a single turn you cannot lose exactly 2 ₪: the expected value describes the mean over time, not one turn.
A fair game

A fair game is one in which the expected gain is 0: over time neither the player nor the organiser gains.

On the wheel of fortune, the mean prize per turn:

50⋅0.1+10⋅0.3+0⋅0.6=5+3=850\cdot0.1+10\cdot0.3+0\cdot0.6=5+3=8
  • If a turn cost 8 ₪, the game would be fair.
  • It costs 10 ₪, so the fair gains 2 ₪ from each turn on average.
An insurance premium and the expected payout

An insurance company sets its premium by the expected value of what it will pay, and adds its profit to it.

Insurance for a mobile phone costs 120 ₪ a year.
The probability that a phone breaks in a year is 0.05\displaystyle 0.05.
When a phone breaks, the company pays 1,500 ₪.

  • The expected payout per customer: 1500⋅0.05+0⋅0.95=75\displaystyle {1500\cdot0.05+0\cdot0.95}={75} ₪.
  • The company’s expected profit from each customer: 120−75=45\displaystyle {120-75}={45} ₪; from 10,000 customers – about 450,000 ₪.
  • For the customer the expected value is negative, −45\displaystyle -45 ₪, and still people take out insurance: they pay a small, known sum so as not to lose a large one all at once.
A raffle ticket

In a raffle, the expected prize is less than the price of a ticket – that is how the prizes and the organiser’s profit are paid for.

A raffle ticket costs 5 ₪.
One ticket in 500 wins 1,000 ₪.
One ticket in 50 wins 20 ₪.

  • The expected prize per ticket: 1000⋅1500+20⋅150=2+0.4=2.4\displaystyle {1000\cdot\frac{1}{500}+20\cdot\frac{1}{50}}={2+0.4}={2.4} ₪.
  • The expected gain: 2.4−5=−2.6\displaystyle {2.4-5}={-2.6} ₪ per ticket.
Exercise 1

At a stall on parents’ day one die is rolled. Taking part costs 4 ₪.
Whoever rolls a 6 wins 15 ₪.
Whoever rolls a 4 or a 5 wins 6 ₪.
Whoever rolls a 1, 2 or 3 wins nothing.

a

Work out the expected prize in one roll.

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

    Work out the expected value of the player’s gain in one roll.

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c

      The parents’ committee wants to raise money for the annual trip. Should it run the stall like this? Explain.

      Exercise 2

      In a café chain’s app a wheel is spun once a day:

      Prize (₪)100200
      Probability0.05\displaystyle 0.050.25\displaystyle 0.250.7\displaystyle 0.7
      a

      Work out the expected prize in one spin.

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        b

        A fair offers the same wheel, and a spin costs 12 ₪. Work out the expected gain per spin.

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Exercise 3

          Insurance against a car’s theft costs 900 ₪ a year.
          The probability that a car is stolen in a year is 0.008\displaystyle 0.008.
          When a car is stolen, the company pays 60,000 ₪.

          a

          Work out the company’s expected payout to one customer in a year.

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            b

            The company has 2,000 customers. How much is it expected to earn in a year?

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              Exercise 4

              In a game two dice are rolled and the numbers added. Taking part costs 4 ₪.
              A sum of 12 wins 70 ₪.
              A sum of 7 wins 10 ₪.
              Any other sum wins nothing.

              Mark in the table the outcomes that win a prize.

              a

              What is the probability of winning any prize?

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                b

                Work out the expected value of the player’s gain in one game.

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  Exercise 5

                  A basketball player scores a three-point shot with probability 0.35\displaystyle 0.35, and a two-point shot with probability 0.5\displaystyle 0.5.

                  a

                  Work out the expected number of points from a three-point shot.

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                    b

                    Work out the expected number of points from a two-point shot.

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                      c

                      Which shot brings the team more points over a whole game? Explain.