Inside the box of every cereal box contains a surprise gift. The store manager assures employees that 23 of the 65 boxes on the shelf have a secret decoder ring and inside the other 42 boxes on the shelf contain a different gift. If two cereal boxes are randomly selected from the shelf to purchase, what is the probability that both of them have the secret decoder ring? (Give answer as a decimal correct to least three decimal places.)

To solve the problem, we need to calculate the probability that both selected cereal boxes contain the secret decoder ring.

### Step 1: Identify the total number of cereal boxes
- Total boxes = 65

### Step 2: Identify the number of boxes with the secret decoder ring
- Boxes with the secret decoder ring = 23

### Step 3: Calculate the probability of selecting the first box with a secret decoder ring
The probability of selecting the first box with a secret decoder ring is:

\[
P(\text{First Box}) = \frac{23}{65}
\]

### Step 4: Calculate the probability of selecting the second box with a secret decoder ring
After selecting the first box with a decoder ring, there are now 22 decoder ring boxes left and 64 boxes in total:

\[
P(\text{Second Box | First Box}) = \frac{22}{64}
\]

### Step 5: Calculate the joint probability that both boxes have a secret decoder ring
To find the probability that both boxes have the decoder ring, multiply the probabilities from Steps 3 and 4:

\[
P(\text{Both Boxes}) = \frac{23}{65} \times \frac{22}{64}
\]

\[
P(\text{Both Boxes}) = \frac{506}{4160} \approx 0.122
\]

Thus, the probability that both selected boxes have a secret decoder ring is **0.122**.

Would you like more details or have any questions? Here are some related problems to consider:

1. What is the probability that neither of the selected boxes has a secret decoder ring?
2. What is the probability that exactly one of the selected boxes has a secret decoder ring?
3. What is the probability that both boxes have the same type of gift?
4. How would the probability change if 3 boxes are selected instead of 2?
5. If 10 more boxes with decoder rings are added to the shelf, what would the new probability be?

**Tip:** When dealing with probabilities involving multiple selections, it's often helpful to calculate the probability step by step, considering how each selection affects the subsequent possibilities.

Calculate the probability of randomly selecting two cereal boxes that both contain a secret decoder ring. Explore key probability concepts, including the multiplication rule for independent events, with a detailed step-by-step solution. Suitable for students in Grades 10-12.

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