Question

The Big River Casino is advertising a new digital lottery-style game called Instant Lotto. The player can win the following monetary prizes with the associated probabilities: 5% probability to win $10 4% probability to win $15 3% probability to win $30 1% probability to win $50 0.1% probability to win the Grand Prize, $1000. (a) Calculate the expected value of the prize for one play of Instant Lotto. (b) As a promotion, a visitor to the casino is given 20 free plays of Instant Lotto. What is the probability that the visitor wins some prize at least twice in the 20 free plays? (c) The number of people who play Instant Lotto each day is approximately normally distributed with a mean of 800 people and a standard deviation of 310 people. What is the probability that a randomly selected day has at least 1000 people play Instant Lotto?

Answer 1

Answer:

(a) The expected value of the prize for one play of Instant Lotto is $3.50.

(b) The probability that the visitor wins some prize at least twice in the 20 free plays is 0.2641.

(c) The probability that a randomly selected day has at least 1000 people play Instant Lotto is 0.2579.

Step-by-step explanation:

(a)

The probability distribution of the monetary prizes that can be won at the game called Instant Lotto is:

X         P (X = x)

$10        0.05

$15        0.04

$30       0.03

$50       0.01

$1000   0.001

$0         0.869

___________

Total =   1.000

Compute the expected value of the prize for one play of Instant Lotto as follows:

$E(X)=\sum x\cdot P (X=x)$

         $=(10\times 0.05)+(15\times 0.04)+(30\times 0.03) \\+ (50\times 0.01)+(1000\times 0.001)+(0\times 0.869)\\=0.5+0.6+0.9+0.5+1+0\\=3.5$          

Thus, the expected value of the prize for one play of Instant Lotto is $3.50.

(b)

Let X = number of times a visitor wins some prize.

A visitor to the casino is given n = 20 free plays of Instant Lotto.

The probability that a visitor wins at any of the 20 free plays is, p = 1/20 = 0.05.

The event of a visitor winning at a random free play is independent of the others.

The random variable X follows Binomial distribution with parameters n = 20 and p = 0.05.

Compute the probability that the visitor wins some prize at least twice in the 20 free plays as follows:

P (X ≥ 2) = 1 - P (X < 2)

              = 1 - P (X = 0) - P (X = 1)

              $=1-[{20\choose 0}0.05^{0}(1-0.05)^{20-0}]-[{20\choose 1}0.05^{1}(1-0.05)^{20-1}]\\=1-0.3585-0.3774\\=0.2641$

Thus, the probability that the visitor wins some prize at least twice in the 20 free plays is 0.2641.

(c)

Let X = number of people who play Instant Lotto each day.

The random variable X is normally distributed with a mean, μ = 800 people and a standard deviation, μ = 310 people.

Compute the probability that a randomly selected day has at least 1000 people play Instant Lotto as follows:

Apply continuity correction:

P (X ≥ 1000) = P (X > 1000 + 0.50)

                    = P (X > 1000.50)

                    $=P(\frac{X-\mu}{\sigma}>\frac{1000.50-800}{310})$

                    $=P(Z>0.65)\\=1-P(Z$

Thus, the probability that a randomly selected day has at least 1000 people play Instant Lotto is 0.2579.