Respuesta :
Answer:
There is a 2.06% probability of winning 15 or more prizes.
Step-by-step explanation:
For each play, there are only two possible outcomes. Either you win a prize, or you do not win a prize. This means that we can solve this problem using concepts of the binomial probability distribution.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinatios of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
In this problem we have that:
There are 20 plays, no [tex]n = 20[/tex]
The chance of winning a prize for each of the 20 plays is 50–50, so [tex]p = 0.50[/tex].
If you bought 20 tickets, what is the chance of winning 15 or more prizes?
This is [tex]P(X \geq 15)[/tex]
[tex]P(X \geq 15) = P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20)[/tex]
[tex]P(X = 15) = C_{20,15}.(0.50)^{15}.(0.50)^{5} = 0.0148[/tex]
[tex]P(X = 16) = C_{20,16}.(0.50)^{16}.(0.50)^{4} = 0.0046[/tex]
[tex]P(X = 17) = C_{20,17}.(0.50)^{17}.(0.50)^{3} = 0.001[/tex]
[tex]P(X = 18) = C_{20,18}.(0.50)^{18}.(0.50)^{2} = 0.0002[/tex]
[tex]P(X = 19) = C_{20,19}.(0.50)^{19}.(0.50)^{1} = 0.00002[/tex]
[tex]P(X = 20) = C_{20,20}.(0.50)^{20}.(0.50)^{0} = 0.00000009[/tex]
So
[tex]P(X \geq 15) = P(X = 15) + P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20) = 0.0148 + 0.0046 + 0.001 + 0.0002 + 0.00002 + 0.00000009 = 0.0206[/tex]
There is a 2.06% probability of winning 15 or more prizes.
