Using the binomial distribution, it is found that there is a [tex]\mathbf{\frac{1}{16}}[/tex] probability you will win the big prize.
For each ball, there are only two possible outcomes, either it is hit or it is not. The probability of hitting a ball is independent of any other ball, 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:
- 4 balls are thrown, hence [tex]n = 4[/tex]
- 0.5 probability of hitting each of them, hence [tex]p = \frac{1}{2}[/tex]
You win the big prize if you hit 4 balls, hence, the probability is P(X = 4).
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 4) = C_{4,4}.\left(\frac{1}{2}\right)^{4}.\left(\frac{1}{2}\right)^{0} = \frac{1}{16}[/tex]
[tex]\mathbf{\frac{1}{16}}[/tex] probability you will win the big prize.
A similar problem is given at https://brainly.com/question/24863377
