Unit 8 · Statistics & ProbabilityS6 · Counting & EV · Telemetry

Expected Value & Multi-Step Probability

Weight each payoff by its probability and combine dependent stages accurately.

28 min
Unlocks: Mission Planner

Ready when you are

7 steps · about 28 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

Finish the previous step to continue, or use “Show everything” above to explore.

Key terms

expected value: The long-run average payoff.
payoff: Net gain or loss for an outcome.
dependent events: Events whose probabilities change after an outcome.

Start by naming the quantity each problem asks for. That habit blocks the most common ACT trap.

Step 2 of 7Try it firstDone

Try it first

Finish the previous step to continue, or use “Show everything” above to explore.

Take a calm first shot. Label what you know, write one relationship, and make the best choice you can.

Step 3 of 7LearnDone

The big idea

Finish the previous step to continue, or use “Show everything” above to explore.

Build the picture before the arithmetic

Include every outcome, including losses and zero. If playing costs money, subtract the fee from each prize or subtract it once from the expected prize. Without replacement, update the denominator.

Key idea

The decision that matters

Expected value is a weighted average: multiply every outcome by its probability, then add.

Expected value

E(X)=\sum_i x_iP(X=x_i)

Write this relationship before inserting numbers. It keeps labels and units attached to the work.

A compact example

A game pays 12 with probability $\frac14$ and 0 otherwise, and costs 2. The expected net is $12\left(\frac14\right)-2=1$ .

Cue	Best move
Several payoffs	multiply each by its probability
Entry fee	subtract once for expected net

Trap

Using the prize instead of the net payoff

This error produces a believable answer. Pause before computing and say what each number means.

Step 4 of 7Worked examplesDone

Worked examples

Finish the previous step to continue, or use “Show everything” above to explore.

Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Expected net value

Problem

A ticket costs 3 dollars. It pays 20 dollars with probability 0.10, 5 dollars with probability 0.30, and nothing otherwise. Find the expected net value.

Expected prize is

20(0.10)+5(0.30)+0(0.60)=3.50

dollars.

Subtract the 3-dollar ticket cost: $3.50-3=0.50$ .

Which calculation gives the expected net value?

The expected net is a gain of 50 cents per play over many plays.

Why are all prizes weighted before subtracting the fee?

Worked example

Example 2: Two stages without replacement

Problem

A bag has 4 red and 6 blue chips. Two chips are drawn without replacement. Find the probability both are red.

The first red probability is

4/10

After one red is removed, the next red probability is $3/9$ .

Which product models two red draws without replacement?

Multiply: $(4/10)(3/9)=2/15$ .

Why do both factors change after the first red draw?

Faded example · you finish it

Example 3: You finish it

Problem

A game pays 12 points with probability 1/4 and 0 otherwise and costs 2 points. Find expected net.

Multiply each payoff by its probability, add, and subtract the cost once.

Which completion is correct?

Which reason validates the result?

Step 5 of 7PracticeDone

Practice

Finish the previous step to continue, or use “Show everything” above to explore.

Step 6 of 7Spot the mistakeDone

Spot the mistake

Finish the previous step to continue, or use “Show everything” above to explore.

Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

Step 7 of 7Exit ticketDone

Exit ticket

Finish the previous step to continue, or use “Show everything” above to explore.

Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.