`{:("Column A" , "In a box of 5 eggs, 2 are rotten","Column B"),( "The probability that one egg chosen ", ,"The probability that two eggs chosen"),( "at random from the box is rotten", ,"at random from the box are rotten"):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to calculate the probabilities for both Column A and Column B based on the given information about the eggs.

### Step 1: Calculate the probability for Column A
Column A states: "The probability that one egg chosen at random from the box is rotten."

- Total eggs = 5
- Rotten eggs = 2

The probability \( P(A) \) that one egg chosen at random is rotten can be calculated using the formula:

\[ P(A) = \frac{\text{Number of rotten eggs}}{\text{Total number of eggs}} = \frac{2}{5} \]

### Step 2: Calculate the probability for Column B
Column B states: "The probability that two eggs chosen at random from the box are rotten."

To find this probability, we need to consider the scenario of choosing 2 rotten eggs from the box of 5 eggs.

1. The total ways to choose 2 eggs from 5 eggs is given by the combination formula \( \binom{n}{r} \):

\[ \text{Total ways to choose 2 eggs from 5} = \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

2. The ways to choose 2 rotten eggs from the 2 rotten eggs available:

\[ \text{Ways to choose 2 rotten eggs from 2} = \binom{2}{2} = 1 \]

3. Now, the probability \( P(B) \) that both eggs chosen are rotten is:

\[ P(B) = \frac{\text{Ways to choose 2 rotten eggs}}{\text{Total ways to choose 2 eggs}} = \frac{1}{10} \]

### Step 3: Compare the probabilities
Now we have:
- Probability from Column A: \( P(A) = \frac{2}{5} = 0.4 \)
- Probability from Column B: \( P(B) = \frac{1}{10} = 0.1 \)

### Step 4: Determine which column is larger
Since \( 0.4 > 0.1 \), we conclude that:

**Column A is larger than Column B.**

### Final Answer
Thus, the answer is that Column A is larger.

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