Question

For a sales​ promotion, the manufacturer places winning symbols under the caps of 27 % of all its soda bottles. If you buy a​ six-pack of​ soda, what is the probability that you win​ something?

Answer 1

0.849

Explanation:

P(winning something)

= 1 - P(winning nothing)

= 1 - (1-0.27)⁶

= 1 - 0.73⁶

= 0.8486657737

Answer 2

84.87% probability that you win​ something

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.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

In this problem we have that:

`n = 6, p = 0.27`

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

`P(X = 0) + P(X \geq 1) = 1`

We want
`P(X \geq 1)`. So

`P(X \geq 1) = 1 - P(X = 0)`

In which

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 0) = C_(6,0).(0.27)^(0).(0.73)^(6) = 0.1513`

`P(X \geq 1) = 1 - P(X = 0) = 1 - 0.1513 = 0.8487`

84.87% probability that you win​ something