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…
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
Answer
Explanation:
Step1: Calculate probabilities
When tossing 2 - 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: Define points and use 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. Let (x_1 = 2) (points for 2 heads), (x_2 = 1) (points for 1 head), (x_3=-3) (points for 0 heads). (E(X)=2\times\frac{1}{4}+1\times\frac{1}{2}+(-3)\times\frac{1}{4})
Step3: Simplify the expression
[ \begin{align*} E(X)&=\frac{2}{4}+\frac{1}{2}-\frac{3}{4}\ &=\frac{2 + 2-3}{4}\ &=\frac{1}{4}=0.25 \end{align*} ]
Answer:
0.25