Question

An unfair coin is flipped. If a head turns up, you win $ 1 . If a tail turns up, you lose $ 1 . The probability of a head is 0.54 and the probability of a tail is 0.46 . Let X be the random variable for the amount won on a single play of this game. What is the expected value of the game?

Answer 1

*Question:-

An unfair coin is flipped. If a head turns up, you win $ 1 . If a tail turns up, you lose $ 1 . The probability of a head is 0.54 and the probability of a tail is 0.46 . Let X be the random variable for the amount won on a single play of this game. What is the expected value of the game?

*Solution:-

E(X) = 1 × 0.63 -1 × 0.37 = 0.26

So we expect to win around 0.26 for each game that we play on this game.

Step-by-step explanation:

For this case we can define a random variable who represent the amount of money win or loss X . X=1 if we got a head and X=-1 if we got a tail.

The expected value of a random variable X is the n-th moment about zero of a probability density function f(x) if X is continuous, or the weighted average for a discrete probability distribution, if X is discrete.

And from definition the expected value is defined with this formula:

$E(X) =\sum_{i=1}^n X_i P(X_i)$

For this case we have the following probabilities:-

• P(X=1) = 0.63,
• P(X=1) = 0.63, P(X=-1) = 0.37

And then we can replace like this:-

E(X) = 1 × 0.63 -1 × 0.37 = 0.26

So we expect to win around 0.26 for each game that we play on this game.