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:
<em>X</em> P (<em>X</em> = <em>x</em>)
$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:
Thus, the expected value of the prize for one play of Instant Lotto is $3.50.
(b)
Let <em>X</em> = number of times a visitor wins some prize.
A visitor to the casino is given <em>n</em> = 20 free plays of Instant Lotto.
The probability that a visitor wins at any of the 20 free plays is, <em>p</em> = 1/20 = 0.05.
The event of a visitor winning at a random free play is independent of the others.
The random variable <em>X</em> follows Binomial distribution with parameters <em>n</em> = 20 and <em>p</em> = 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](https://tex.z-dn.net/?f=%3D1-%5B%7B20%5Cchoose%200%7D0.05%5E%7B0%7D%281-0.05%29%5E%7B20-0%7D%5D-%5B%7B20%5Cchoose%201%7D0.05%5E%7B1%7D%281-0.05%29%5E%7B20-1%7D%5D%5C%5C%3D1-0.3585-0.3774%5C%5C%3D0.2641)
Thus, the probability that the visitor wins some prize at least twice in the 20 free plays is 0.2641.
(c)
Let <em>X</em> = number of people who play Instant Lotto each day.
The random variable <em>X</em> is normally distributed with a mean, <em>μ</em> = 800 people and a standard deviation, <em>μ</em> = 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)
Thus, the probability that a randomly selected day has at least 1000 people play Instant Lotto is 0.2579.
