NCEA Level 2Expected valueAS 91267

Expected value and expected numbers

What this lesson covers

  • An expected number is the probability multiplied by the number of trials.
  • Expected value is a long-run average, so it need not be a possible single outcome.
Key idea

What to expect

  • A probability says how likely.
  • An expected number says how many.
  • An expected value says how much, on average.
Worked example

Expected number: n × p

  • 11% buy a warranty.
  • Expected number = n × p.
  1. n = 400, p = 0.11
  2. expected = n × p
  3. = 400 × 0.11
  4. = 44 customers
Key idea

Expected is not guaranteed

  • Another 400 customers might give 40, or 49.
  • 44 is the long-run average.
  • Say "about 44" or "expected".
Worked example

A game: $2.50 to roll a die

  • A 6 wins $9.
  • A 4 or 5 wins $2.
  • Anything else wins nothing.
OutcomePWin
Roll a 61/6$9
Roll a 4 or 52/6$2
Roll 1, 2 or 33/6$0
Worked example

The expected value of the game

  • Σ prize × probability.
  • Then subtract the cost.
  1. E(win) = 9 × 1/6 + 2 × 2/6
  2. + 0 × 3/6
  3. = 1.5 + 0.6667 + 0
  4. = $2.1667
  5. E(net) = 2.1667 − 2.50subtract the cost
  6. = −$0.3333about −33c a game
Key idea

You never lose exactly 33 cents

  • No game loses exactly 33c.
  • A long-run average.
  1. roll 6: 9 − 2.50 = $6.50
  2. roll 4, 5: 2 − 2.50 = −$0.50
  3. roll 1–3: 0 − 2.50 = −$2.50
  4. E(net) = 6.50 × 1/6 − 0.50 × 2/6
  5. − 2.50 × 3/6
  6. = 1.0833 − 0.1667 − 1.25
  7. = −$0.33as before
Key idea

Is it a fair game?

  • Fair means E(net) = 0.
  • Fee: E(win), to the cent.
  1. fair: E(net) = 0
  2. E(win) − fee = 0
  3. fee = $2.1667
  4. ≈ $2.17to the cent
  5. 2.1667 − 2.17 = −0.0033a third of a cent
Worked example

Expected numbers from a tree

  • p comes from the tree.
  • Round to whole people.
  1. P(warranty) = 0.11from the tree
  2. expected = 250 × 0.11
  3. = 27.5
  4. about 28 customerspeople are whole
Key idea

Round in context

  • People and days are whole numbers.
  • Money is to the cent.
  • Keep full accuracy until the final answer.
ValueBecauseReport
27.5 customerspeople are wholeabout 28
2.64 late daysdays are wholeabout 3
$2.1667 per gamemoney$2.17
Key idea

Where the marks are

  • Achieved: an expected number.
  • Merit: an expected value using a tree.
  • Excellence: interpreting it, and its limits.
Summary

What to take away

  • n × p for numbers; Σ value × p for value.
Expected numbern × p
Expected valueΣ value × probability
Netsubtract the cost
Meaninga long-run average
Roundingwhole things in context, money to cents
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • Question 2: subtract the ticket price.
  • Round sensibly in context.
Question
1P(defect) = 0.035. How many defects in 2000 items?
2500 tickets at $5, prizes $500 and $200. E(net)?
3P(late) = 0.12 a day. Late days in 22 days?
4P(claim) 0.02, pays $3000, premium $80. Profit?
Show answer

1. 0.035 × 2000 = 70. 2. E(win) = 700 ÷ 500 = $1.40, so E(net) = 1.40 − 5 = −$3.60. 3. 0.12 × 22 = 2.64, about 3 days. 4. Expected payout 0.02 × 3000 = $60, so $80 − $60 = $20 expected profit per policy.

Worked example

The answers: questions 1 and 2

  1. 1. expected = 2000 × 0.035
  2. = 70 defects
  3. 2. P(each prize) = 1/500
  4. E(win) = 500 × 1/500
  5. + 200 × 1/500
  6. = 1 + 0.40 = $1.40
  7. E(net) = 1.40 − 5 = −$3.60per ticket
Worked example

The answers: questions 3 and 4

  1. 3. expected = 22 × 0.12
  2. = 2.64
  3. about 3 daysdays are whole
  4. 4. E(payout) = 3000 × 0.02
  5. = $60
  6. profit = 80 − 60 = $20per policy

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