Expected Value Statistics Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

medium
A game costs \2 to play. You roll a fair die: if you roll a 6, you win \10; otherwise, you win nothing. Find the expected value per game.

Solution

  1. 1
    Step 1: P(6)=16P(6) = \frac{1}{6}, winnings = $10โˆ’$2=$8\$10 - \$2 = \$8 net. P(notย 6)=56P(\text{not 6}) = \frac{5}{6}, winnings = $0โˆ’$2=โˆ’$2\$0 - \$2 = -\$2 net.
  2. 2
    Step 2: E(X)=16(8)+56(โˆ’2)=86โˆ’106=โˆ’26โ‰ˆโˆ’$0.33E(X) = \frac{1}{6}(8) + \frac{5}{6}(-2) = \frac{8}{6} - \frac{10}{6} = -\frac{2}{6} \approx -\$0.33.
  3. 3
    Step 3: On average, you lose about 33 cents per game.

Answer

E(X)โ‰ˆโˆ’$0.33E(X) \approx -\$0.33 per game.
Expected value is the long-run average outcome. A negative expected value means the game favours the house โ€” on average, you lose money over many plays.

About Expected Value

The expected value of a random variable is the long-run average outcome of a random process, calculated as the weighted sum of each possible outcome times its probability. It represents what you would earn or lose on average per trial if the process were repeated infinitely many times.

Learn more about Expected Value โ†’

More Expected Value Examples

Example 2 medium

A raffle sells 200 tickets at [formula]500. Find the expected value for a ticket buyer.

Example 3 medium

A spinner has outcomes: [formula]3 (prob 0.3), $10 (prob 0.2). Find the expected value.

Example 4 medium

A game costs [formula]20 with probability 0.1, [formula]0 otherwise. Find the expected net value of

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