Probability · Ordinary Level
Expected Value
Ordinary Level · Class 10 · Tap NEXT to begin
Section 1 of 2
Expected Value
Average winnings
The expected value is the average amount per play: $E(x) = \sum x \cdot P(x)$ — each prize times its probability, all added. Then subtract the cost to see if the game is worth playing.
Worked example — an even spinner
A spinner has $4$ equal sections: €$5$, €$2$, €$20$, €$10$. It costs €$8$ to play. What do you expect to win per play?
Average win $= \dfrac{5+2+20+10}{4} = \dfrac{37}{4} = $ €$9.25$
Net $= 9.25 - 8 = $ €$1.25$ per play
win €9.25; net €1.25
Worked example — an uneven spinner
A spinner shows €$0$ (two sections), €$10$ and €$5$. It costs €$4$ to play. Find the expected value.
$E(x) = \tfrac{1}{2}(0) + \tfrac{1}{4}(10) + \tfrac{1}{4}(5) = 0 + 5 + 1.25 = $ €$6.25$
Net $= 6.25 - 4 = $ €$2.25$
$E(x) = $ €6.25; net €2.25
Worked example — three cards
Three cards $A, B, C$: $A$ worth €$50$, $B$ €$20$, $C$ €$5$. Pick one at random; it costs €$30$ to play. Find the expected value.
$E(x) = \tfrac{1}{3}(50) + \tfrac{1}{3}(20) + \tfrac{1}{3}(5) = \tfrac{75}{3} = 25$
Net $= 25 - 30 = -$€$5$ (a losing game)
$E(x) = $ €25; net $-$€5
You try
A bag holds €$2$, €$1$, €$2$, €$2$, €$1$, €$5$. You pick one; it costs €$2.50$ to play. Find the expected value and the net result.
$E(x) = \tfrac{1}{2}(2) + \tfrac{1}{3}(1) + \tfrac{1}{6}(5)$ (three €2’s, two €1’s, one €5).
$E(x) = \tfrac{1}{2}(2) + \tfrac{1}{3}(1) + \tfrac{1}{6}(5) = 1 + 0.33 + 0.83 = $ €$2.17$
Net $= 2.17 - 2.50 = -$€$0.33$
$E(x) = $ €2.17; net $-$€0.33
That’s Class 10.
Expected value $E(x) = \sum x\cdot P(x)$, and whether a game is worth the cost. Class 11: permutations.