91Ó°ÊÓ

The number of moons of all the 8 planets is listed.

a.

The total number of possible sequences is computed below:

\(10 \times 10 \times 10 = 1000\)

The number of winning sequences = 1

The probability of winning is computed below:

\(\begin{array}{c}P\left( {{\rm{winning}}} \right) = \frac{{{\rm{Number}}\;{\rm{of}}\;{\rm{winning}}\;{\rm{sequences}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{sequences}}}}\\ = \frac{1}{{1000}}\\ = 0.001\end{array}\)

Therefore, the probability of winning is equal to 0.001.

b.

The number of days in a year is equal to n=365.

The probability of winning a game is equal to p=0.001.

The mean number of wins in a year if one game is played every day is computed below:

\(\begin{array}{c}\mu = np\\ = 365\left( {0.001} \right)\\ = 0.365\end{array}\)

Thus, the mean number of wins in a year if one game is played every day is equal to 0.365.

c.

Let X denote the number of wins in a year.

So, X follows a Poisson distribution with mean equal to 0.365.

The probability of winning once in a year is computed below:

\[\begin{array}{c}P\left( {X = x} \right) = \frac{{{\mu ^x}{e^{ - \mu }}}}{{x!}}\\P\left( {X = 1} \right) = \frac{{{{\left( {0.365} \right)}^1}{{\left( {2.71828} \right)}^{ - 0.365}}}}{{1!}}\\ = 0.253\end{array}\]\(\left( {{x_{{\rm{loss}}}}} \right)\)

Thus, the probability of winning once in a year is equal to 0.253.

d.

The amount received on winning\(\left( {{x_{{\rm{win}}}}} \right)\)is equal to $499.

The probability of winning\(\left( {{p_{win}}} \right)\)is equal to 0.001.

The amount lost on losing is equal to -1dollar.

The probability of losing is equal to:

\(\begin{array}{c}{p_{{\rm{loss}}}} = 1 - {p_{win}}\\ = 1 - 0.001\\ = 0.999\end{array}\)

The expected value lost/gained on the purchase of one ticket is computed below:

\(\begin{array}{c}E = {x_{win}}{p_{win}} + {x_{loss}}{p_{loss}}\\ = \left( {499} \right)\left( {0.001} \right) + \left( { - 1} \right)\left( {0.999} \right)\\ = - 0.50\end{array}\)

Thus, the expected value for the purchase of one ticket is equal to – 0.50 dollars or – 50 cents.