Answer:
84.87% probability that you win something
Step-by-step explanation:
For each bottle of soda, there are only two possible outcomes. Either there is a winning symbol, or there is not. For each bottle, the probability of having a winning symbol is independent from other bottles. So we use the binomial probability distribution to solve this problem.
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 combinations 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:
[tex]n = 6, p = 0.27[/tex]
If you buy a six-pack of soda, what is the probability that you win something?
You will will something if at least one of the bottles has a prize.
We know that either no bottles have a prize, or at least one does. So
[tex]P(X = 0) + P(X \geq 1) = 1[/tex]
We want [tex]P(X \geq 1)[/tex]. So
[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{6,0}.(0.27)^{0}.(0.73)^{6} = 0.1513[/tex]
[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.1513 = 0.8487[/tex]
84.87% probability that you win something
Answer:
0.849
Step-by-step explanation:
P(winning something)
= 1 - P(winning nothing)
= 1 - (1-0.27)⁶
= 1 - 0.73⁶
= 0.8486657737
