Out of the cards 5, 5, 6, 7, 9, if two cards are picked consecutively, what is the probability
that the number on the 2nd card is higher than the number on the 1st card?

We are given the cards: 5, 5, 6, 7, 9. Two cards are picked consecutively, and we are asked to find the probability that the number on the second card is higher than the number on the first card.

### Step 1: Total Possible Outcomes
There are 5 cards in total, and two cards are picked consecutively, meaning order matters. The total number of possible outcomes for picking two cards is:

\[
\text{Total outcomes} = 5 \times 4 = 20
\]

### Step 2: Successful Outcomes
Now, let's count the number of favorable outcomes where the second card is higher than the first card. We can list out all the possible outcomes where the number on the second card is higher than the number on the first card:

- If the first card is **5**:
  - The second card can be 6, 7, or 9.
  - There are 3 outcomes: (5, 6), (5, 7), (5, 9)
- If the first card is **5** (the second 5):
  - The second card can still be 6, 7, or 9.
  - There are 3 outcomes: (5, 6), (5, 7), (5, 9)
- If the first card is **6**:
  - The second card can be 7 or 9.
  - There are 2 outcomes: (6, 7), (6, 9)
- If the first card is **7**:
  - The second card can only be 9.
  - There is 1 outcome: (7, 9)
- If the first card is **9**:
  - There is no card higher than 9, so there are no outcomes.

### Step 3: Count of Successful Outcomes
Adding up all the successful outcomes:

\[
3 \ (\text{from first 5}) + 3 \ (\text{from second 5}) + 2 \ (\text{from 6}) + 1 \ (\text{from 7}) = 9 \text{ successful outcomes}
\]

### Step 4: Probability Calculation
The probability that the number on the second card is higher than the number on the first card is:

\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{9}{20}
\]

Thus, the probability is:

\[
\boxed{\frac{9}{20}}
\]

Would you like further details, or have any additional questions?

Here are 5 related questions to explore further:
1. What if the cards had been replaced after picking?
2. How does the probability change

Out of the cards 5, 5, 6, 7, 9, if two cards are picked consecutively, what is the probability that the number on the 2nd card is higher than the number on the 1st card?

Explore how to calculate the probability of picking two consecutive cards from the set 5, 5, 6, 7, 9 where the second card has a higher value than the first. Learn key concepts in probability and combinatorics, with a detailed step-by-step solution, suitable for Grades 9-10.

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