`{:("Column A","Everyone who passes the test will be awarded a degree. The probability that Tom passes the test is 0.5, and the probability that John passes the test is 0.4. The two events are independet of each other.", "Column B"),("The probability that both Tom and John get the degree",,"The probability that at least one of them gets the degree"):}`

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 columns A and B as described in the question.

### Step 1: Understand the Problem
We have two individuals, Tom and John, who have probabilities of passing a test given as:
- Probability that Tom passes (P(Tom)) = 0.5
- Probability that John passes (P(John)) = 0.4

The events of passing are independent of each other.

### Step 2: Calculate the Probability that Both Tom and John Get the Degree (Column A)
To find the probability that both Tom and John pass the test, we use the formula for independent events:
\[ P(A \cap B) = P(A) \times P(B) \]
Where:
- \( P(A) \) is the probability that Tom passes = 0.5
- \( P(B) \) is the probability that John passes = 0.4

Substituting the values:
\[ P(Tom \text{ and } John) = P(Tom) \times P(John) = 0.5 \times 0.4 = 0.2 \]

Thus, the probability that both Tom and John get the degree (Column A) is **0.2**.

### Step 3: Calculate the Probability that At Least One of Them Gets the Degree (Column B)
To find the probability that at least one of them passes, we use the formula for the union of two independent events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Where:
- \( P(A) \) = 0.5 (Tom passing)
- \( P(B) \) = 0.4 (John passing)
- \( P(A \cap B) \) = 0.2 (both passing, calculated in Step 2)

Substituting the values:
\[ P(Tom \text{ or } John) = P(Tom) + P(John) - P(Tom \text{ and } John) \]
\[ P(Tom \text{ or } John) = 0.5 + 0.4 - 0.2 = 0.7 \]

Thus, the probability that at least one of them gets the degree (Column B) is **0.7**.

### Step 4: Compare the Results
Now we compare the results from Column A and Column B:
- Column A: 0.2
- Column B: 0.7

Since **0.7 > 0.2**, we conclude that Column B is larger than Column A.

### Final Answer
The answer is that **Column B is larger**.

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