Respuesta :
Answer:
The expectation (in dollars) for this game is -$0.50.
Step-by-step explanation:
The sample space of flipping two fair coins together is:
S = {HH, HT, TH, TT}
Here,
H = heads comes up
T = tails comes up
Each event in the sample space is equally likely, i.e.
P (HH) = P( HT) = P (TH) = P (TT) = 0.25
The conditions of the game are:
- If both coins come up tails, the person flipping the coins looses $6.
- If one shows heads and one shows tails, the person flipping the coins gains $2.
- If both coins come up heads, its a draw.
The probability distribution is as follows:
Outcomes X P (X)
HH $0 0.25
HT $2 0.25
TH $2 0.25
TT -$6 0.25
The expected value of the probability distribution is given by the formula:
[tex]E(X)=\sum x\cdot P (X)[/tex]
Compute the expected value of the game as follows:
[tex]E(X)=\sum x\cdot P (X)[/tex]
[tex]=(\$0\times 0.25)+(\$2\times 0.25)+(\$2\times 0.25)+(-\$6\times 0.25)\\=\$0+\$0.50+\$0.50-\$1.50\\=-\$0.50[/tex]
Thus, the expectation (in dollars) for this game is -$0.50.
Using the binomial distribution and the expected value, it is found that the expectation for this game is of $0.5.
For each toss, there are only two possible outcomes, either it is heads or it is tails. The outcome of a toss is independent of any other toss, which means that the binomial distribution is used to solve this question.
Binomial probability distribution
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
- x is the number of successes.
- n is the number of trials.
- p is the probability of a success on a single trial.
In this problem:
- Fair coin, that is, equally as likely to be heads or tails, hence [tex]p = 0.5[/tex].
- Two coins are flipped, hence [tex]n = 2[/tex]
The probabilities are:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{2,0}.(0.5)^{0}.(0.5)^{2} = 0.25[/tex]
[tex]P(X = 1) = C_{2,1}.(0.5)^{1}.(0.5)^{1} = 0.5[/tex]
[tex]P(X = 2) = C_{2,2}.(0.5)^{2}.(0.5)^{0} = 0.25[/tex]
The expected value is given by the sum of each outcome multiplied by it's probability, hence:
[tex]E(X) = 0.5(-2) + 0.25(6) = 0.5[/tex]
The expectation for this game is of $0.5.
A similar problem is given at https://brainly.com/question/24855677
