Question

There are 6 cards numbered 1 to 6, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two cards drawn. Then P(X > 3) is:

• A. \(\frac{14}{15}\)
• B. \(\frac{1}{15}\)
• C. \(\frac{11}{12}\)
• D. \(\frac{1}{12}\)

Hint:

In problems involving combinations and probability, it's important to first calculate the total number of possible outcomes and then determine the number of favorable outcomes. To find the probability, use the ratio of favorable outcomes to total outcomes. In this case, identifying the unfavorable outcomes (those where the sum is less than or equal to 3) simplifies the problem and leads to a straightforward calculation of the probability.

Answer 1

The Correct Option is A

The total number of ways to choose 2 cards from 6 is:
\(\binom{6}{2} = 15.\)
The event \(X > 3\) means the sum of the numbers on the two cards is greater than 3. The only pair with a sum \(\leq 3\) is \((1, 2)\), which occurs in 1 way.

Thus, the number of favorable outcomes for \(X > 3\) is:
\(15 - 1 = 14.\)

The probability is:
\(P(X > 3) = \frac{14}{15}.\)

Answer 2

The total number of ways to choose 2 cards from 6 is:

Using the combination formula, the number of ways to choose 2 cards from 6 is given by: \[ \binom{6}{2} = \frac{6!}{2!(6 - 2)!} = \frac{6 \times 5}{2 \times 1} = 15. \]

Step 1: Identify the event \( X > 3 \):

The event \( X > 3 \) means the sum of the numbers on the two cards is greater than 3. The only pair with a sum less than or equal to 3 is \( (1, 2) \), which occurs in 1 way.

Step 2: Calculate the number of favorable outcomes:

The total number of outcomes is 15, and the only unfavorable outcome is the pair \( (1, 2) \), which occurs in 1 way. Therefore, the number of favorable outcomes for \( X > 3 \) is: \[ 15 - 1 = 14. \]

Step 3: Calculate the probability:

The probability of the event \( X > 3 \) is the ratio of favorable outcomes to total outcomes: \[ P(X > 3) = \frac{14}{15}. \]

Conclusion: The probability that the sum of the numbers on the two cards is greater than 3 is \( \frac{14}{15} \).