To calculate the probability that the number on the second card Helen takes is greater than the number on the first card she takes, follow these steps:
1. Identify the Total Number of Cards and Possible Pairs:
- We have 7 cards numbered 1 through 7.
- Helen can choose the first and the second cards in different possible orders.
- Since Helen does not replace the card after the first draw, the number of possible pairs [tex]\((i, j)\)[/tex] where [tex]\(i \neq j\)[/tex] is [tex]\(7 \times 6 = 42\)[/tex] pairs (because she has 7 choices initially and 6 choices for the second draw).
2. Count the Favorable Pairs:
- To count the favorable pairs where the number on the second card is greater than the number on the first card, we need to consider pairs [tex]\((i, j)\)[/tex] with [tex]\(i \neq j\)[/tex] such that [tex]\(j > i\)[/tex].
- For example:
- If Helen picks 1 first, the favorable second picks are 2, 3, 4, 5, 6, and 7 (6 options).
- If Helen picks 2 first, the favorable second picks are 3, 4, 5, 6, and 7 (5 options), and so on.
- Summing these, we get:
[tex]\[
6 + 5 + 4 + 3 + 2 + 1 = 21 \text{ favorable pairs}
\][/tex]
3. Calculate the Probability:
- The probability is the ratio of favorable pairs to total pairs.
- Using the counts we have:
[tex]\[
\text{Probability} = \frac{\text{Favorable Pairs}}{\text{Total Pairs}} = \frac{21}{42} = 0.5
\][/tex]
Therefore, the probability that the number on the second card Helen takes is greater than the number on the first card she takes is [tex]\(0.5\)[/tex] or 50%.
