
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?

To solve this problem, we'll calculate the probability of winning at least one prize when buying a six-pack of soda bottles, given that each bottle has a 27% chance of containing a winning symbol.

### Step 1: Define the probability of not winning.
- The probability of **not winning** on a single bottle is the complement of the probability of winning:
  \[
  P(\text{Not Winning}) = 1 - 0.27 = 0.73
  \]

### Step 2: Calculate the probability of not winning on all six bottles.
- The probability of not winning on all six bottles is:
  \[
  P(\text{Not Winning on all six bottles}) = 0.73^6
  \]

### Step 3: Calculate the probability of winning at least one prize.
- The probability of winning at least one prize is the complement of the probability of not winning on all six bottles:
  \[
  P(\text{Winning at least one prize}) = 1 - P(\text{Not Winning on all six bottles})
  \]
  \[
  P(\text{Winning at least one prize}) = 1 - 0.73^6
  \]

Now, let's compute the value of \( 0.73^6 \).

\[
0.73^6 \approx 0.2325
\]

Thus,

\[
P(\text{Winning at least one prize}) \approx 1 - 0.2325 = 0.7675
\]

### Final Answer:
The probability that you win something when you buy a six-pack of soda is approximately **0.7675**, or **76.75%**.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. What is the probability of winning exactly one prize in the six-pack?
2. How would the probability change if the chance of winning under each cap increased to 35%?
3. If you bought two six-packs, what is the probability of winning at least one prize?
4. What is the expected number of winning caps in a six-pack?
5. How does the probability change if the number of bottles in the pack is increased to 12?

**Tip:** When solving problems involving probabilities of multiple independent events, remember to use the complement rule if calculating the probability of "at least one" occurrence.

Learn how to calculate the probability of winning something in a sales promotion where 27% of soda bottles have winning symbols under their caps. Understand the complement rule for probability and its application. Suitable for Grades 9-12 students.

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