Answer:
The probability that you win at least $1 both times is 0.25 = 25%.
Step-by-step explanation:
For each game, there are only two possible outcomes. Either you win at least $1, or you do not. Games are independent. This means that the binomial probability distribution is used to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
In which
is the number of different combinations of x objects from a set of n elements, given by the following formula.
![C_{n,x} = \frac{n!}{x!(n-x)!}](https://tex.z-dn.net/?f=C_%7Bn%2Cx%7D%20%3D%20%5Cfrac%7Bn%21%7D%7Bx%21%28n-x%29%21%7D)
And p is the probability of X happening.
Probability of winning at least $1 on a single game:
The die has 6 sides.
If it lands on 4, 5 or 6(either of the three sides), you win at least $1. So
![p = \frac{1}{2} = 0.5](https://tex.z-dn.net/?f=p%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%3D%200.5)
You are going to play the game twice.
This means that ![n = 2](https://tex.z-dn.net/?f=n%20%3D%202)
The probability that you win at least $1 both times is
This is P(X = 2).
![P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}](https://tex.z-dn.net/?f=P%28X%20%3D%20x%29%20%3D%20C_%7Bn%2Cx%7D.p%5E%7Bx%7D.%281-p%29%5E%7Bn-x%7D)
![P(X = 2) = C_{2,2}.(0.5)^{2}.(0.5)^{2} = 0.25](https://tex.z-dn.net/?f=P%28X%20%3D%202%29%20%3D%20C_%7B2%2C2%7D.%280.5%29%5E%7B2%7D.%280.5%29%5E%7B2%7D%20%3D%200.25)
The probability that you win at least $1 both times is 0.25 = 25%.