in each turn of a game you toss two coins. if 2 heads come up, you win 2 points and if 1 head comes up you win 1 point. if no heads come up, you lose 3 points.
what is the expected value of the number of points for each turn?
this game is

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Accepted Answer

# Explanation:
## Step1: Calculate the probability of each outcome
When tossing two coins, the sample space is $S=\{HH, HT, TH, TT\}$, $n(S) = 4$.
- Probability of 2 heads ($P(2H)$): $P(2H)=\frac{1}{4}$ (only $HH$ case).
- Probability of 1 head ($P(1H)$): $P(1H)=\frac{2}{4}=\frac{1}{2}$ ($HT$ and $TH$ cases).
- Probability of 0 heads ($P(0H)$): $P(0H)=\frac{1}{4}$ (only $TT$ case).

## Step2: Use the expected - value formula $E(X)=\sum_{i}x_{i}P(x_{i})$
Let $x_1 = 2$ (points for 2 heads), $x_2=1$ (points for 1 head), $x_3=- 3$ (points for 0 heads).
$$
\begin{align*}
E(X)&=(2)\times\frac{1}{4}+(1)\times\frac{1}{2}+(-3)\times\frac{1}{4}\
&=\frac{2}{4}+\frac{1}{2}-\frac{3}{4}\
&=\frac{2 + 2-3}{4}\
&=\frac{1}{4}=0.25
\end{align*}
$$

# Answer:
The expected value of the number of points for each turn is $0.25$. Since the expected value $E(X)=0.25>0$, this game is fair (if $E(X) = 0$) or unfair in favor of the player (since $E(X)>0$).

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Question

Explanation:

Step1: Calculate the probability of each outcome

Step2: Use the expected - value formula \(E(X)=\sum_{i}x_{i}P(x_{i})\)

Answer: