Expected value in the context of a coin flip represents the average outcome you would expect over a large number of flips.
Here’s how to understand the expected value for a coin flip:
- Outcome 1: Heads (H) Let’s assign a numerical value of 1 to heads, representing a “success” or a gain.
- Outcome 2: Tails (T) Let’s assign a numerical value of 0 to tails, representing a “failure” or no gain.
For a fair coin , the probability of getting heads is 0.5 (or 1/2), and the probability of getting tails is also 0.5 (or 1/2).
The formula for expected value (E(X)) is the sum of each outcome (x) multiplied by its probability (P(x)): E(X) = Σ (x * P(x)).
Applying this to a single fair coin flip:
E(X) = (Outcome of Heads Probability of Heads) + (Outcome of Tails Probability of Tails)
E(X) = (1 0.5) + (0 0.5) = 0.5 + 0 = 0.5.
The expected value of a single fair coin flip, when heads is assigned a value of 1 and tails 0, is 0.5. This doesn’t mean you will get half a head on any single flip, but rather that if you flip the coin many times, the average number of heads will approach 0.5 per flip.
For example
- If you flip a fair coin 100 times, the expected number of heads is 50, according to [www.vaia.com].
- If a game involves winning $2 for heads and losing $1 for tails, the expected value of playing the game with 1000 flips is $500, according to [Mathematics Stack Exchange].
Important Note: Some studies suggest that real-world coin flips may not be perfectly 50/50, potentially favoring the side that starts face up slightly, possibly around 51/49. However, in most theoretical and statistical contexts, a fair coin with a 50/50 probability is assumed.
Is flipping a coin actually 50/50?
Good point! He said his team found that coins flipped into the air and caught in the hand had a 50.8% chance of landing on the same side they started from.
What is the paradox of coin flip expected value?
The St. Petersburg paradox or St. Petersburg lottery is a paradox involving the game of flipping a coin where the expected payoff of the lottery game is infinite but nevertheless seems to be worth only a very small amount to the participants.
Why did we not get 50% heads and 50% tails?
In 2007, researchers theorised that when a coin is flipped, the flipper’s thumb imparts a slight wobble to it, causing it to spend more time with one side facing upwards while in the air and making it more likely to land showing that side.
Do coin flips predict the future?
For example, if a coin has landed on heads five times in a row, some people believe that tails is more likely to come up next. But that’s the Gambler’s Fallacy. In reality, each coin toss is an independent event and the previous outcomes have no bearing on future ones.