identifying a fair game\nin each turn of a game you toss two coins. if 2 heads come up, you win 2 points and…
identifying a fair game\nin 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.\nwhat is the expected value of the number of points for each turn?\n-0.25\n0\n0.25\n0.5\ndone
Answer
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