Question

identifying a fair game
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?
-0.25
0
0.25
0.5
done

Explanation:

Step1: Calculate probabilities of coin - toss outcomes

When tossing two coins, the sample space is \(S=\{HH, HT, TH, TT\}\), \(n(S) = 4\). The probability of getting 2 heads \(P(2H)=\frac{1}{4}\), the probability of getting 1 head \(P(1H)=\frac{2}{4}=\frac{1}{2}\), and the probability of getting 0 heads \(P(0H)=\frac{1}{4}\).

Step2: Calculate the expected - value formula

The expected - value formula is \(E(X)=\sum_{i}x_{i}P(x_{i})\), where \(x_{i}\) are the possible values and \(P(x_{i})\) are their corresponding probabilities. Here, \(x_1 = 2\) (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:

0.25