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Expected Value with Tables, Binomials, and Areas

·583 words·3 mins
Author
Adrian

What Is Expected Value?
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Expected value (EV) tells you, on average, what to expect in a probabilistic setting — like gambling, games, or life.

Definition: Expected value is the weighted average of all outcomes, where each outcome is multiplied by its probability.

For a basic example:

  • Heads: earn \( \$5 \)
  • Tails: lose \( \$10 \)
  • Both have \( \frac{1}{2} \) probability
\[ \text{EV} = \frac{1}{2} \cdot 5 + \frac{1}{2} \cdot (-10) = -2.5 \]

So you lose $2.50 per flip on average: not a fair game.

Expected Value in the Lottery
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Let’s say you analyze the Florida Lottery with outcomes and probabilities. If the weighted sum of your winnings is:

\[ \text{Total EV} = 68 \]

You spent $2 on the ticket, so your net expected value is:

\[ 68 - 2 = 66 \]

But suppose the real math gives:

\[ \text{EV} = -1.33 \]

This means you’re losing money in the long run. Even with huge jackpots, the odds are against you.

Word Rearrangement (Counting Methods)
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Say you’re arranging letters:

  • For the word ADRI: All letters are unique. Total permutations:
\[ 4! = 24 \]
  • For the word ADRIAN: Repeated letters must be adjusted:
\[ \frac{6!}{2!} = 360 \]
  • Lastly, for the word SKASKA: Repeated letters must be adjusted again:
\[ \frac{6!}{2! \cdot 2! \cdot 2!} = 90 \]

True/False Chemistry Test
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You take a 10-question true/false test randomly. What’s the chance of scoring exactly 7 correct?

This is a binomial probability problem:

\[ P = \frac{10!}{7! \cdot 3!} \cdot \left( \frac{1}{2} \right)^{10} = \frac{120}{1024} \approx 11.7\% \]

Area-Based Probability
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You shoot a basketball at random on a square backboard of area \( 10 \times 10 = 100 \). The hoop is a circle of radius 3:

\[ \text{Success Area} = \pi \cdot 3^2 = 9\pi \approx 28.27 \]

\[ P(\text{make}) = \frac{9\pi}{100} \approx 0.28 \]

Bet 1: Million-Dollar Target
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You pay $10,000 to shoot a random point at a 25×30 rectangle containing a small diamond-shaped region (formed by joining midpoints of a 2×3 rectangle). Only 2 of 8 internal triangles are shaded:

\[ \text{Success Area} = \frac{1}{4} \cdot 6 = 1.5 \]

\[ \text{Total Area} = 25 \cdot 30 = 750 \]

\[ \text{EV} = \frac{1.5}{750} \cdot 1{,}000{,}000 - 10{,}000 = -\$8{,}000 \]

Don’t take this bet.

Bet 2: Deck of Cards
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  • Draw a card.
  • Win $35,000 if it’s a face card or ace (total of 16 cards).
  • Pay $10,000 to play.
\[ \text{P(win)} = \frac{16}{52},\quad \text{EV} = \frac{16}{52} \cdot 35{,}000 - 10{,}000 = -\$2{,}230.77 \]

Again, not worth it.

Bet 3: Scrabble Tiles
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You draw 3 Scrabble tiles with replacement from a bag of 60 (18 vowels, 42 consonants). Payouts:

  • 0 vowels → $0
  • 1 vowel → $20,000
  • 2 vowels → $20,000
  • 3 vowels → Lose $90,000

Let’s compute:

  • \( P(0) = \left( \frac{42}{60} \right)^3 \)
  • \( P(1) = 3 \cdot \left( \frac{18}{60} \right) \cdot \left( \frac{42}{60} \right)^2 \)
  • \( P(2) = 3 \cdot \left( \frac{18}{60} \right)^2 \cdot \left( \frac{42}{60} \right) \)
  • \( P(3) = \left( \frac{18}{60} \right)^3 \)
\[ \text{EV} = P(1) \cdot 20{,}000 + P(2) \cdot 20{,}000 - P(3) \cdot 90{,}000 - 10{,}000 \approx +\$170 \]

This one is actually favorable. Take the bet!

Summary
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Bet/Game Expected Value Should You Take It?
Coin Flip -$2.50 Nope
Lottery Ticket -$1.33 Nope
Target Shooting -$8,000 Nope
Card Bet -$2,230 Nope
Scrabble Bet +$170 Go ahead

Credits
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All explanations and problems written by Adrian Hernandez Vega, unless otherwise noted.

Prefer video? Watch the full explanation here: