Question
A bag contains \( n \) cards, each having one of the numbers \( (1,2,3, \ldots, n) \) written on it. If all numbers are used, the probability of drawing a card with a number less than or equal to 5 is \( \frac{1}{10} \). How many cards are in the bag? There are \( \square \) cards in the bag.
Ask by Goodwin Powers. in the United States
Jan 14,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
There are 50 cards in the bag.
Solution
To determine the number of cards in the bag, let's analyze the given information:
1. **Total Number of Cards:** Let the total number of cards be \( n \).
2. **Favorable Outcomes:** The numbers on the cards range from 1 to \( n \). The numbers less than or equal to 5 are \( 1, 2, 3, 4, 5 \). So, there are **5** favorable outcomes.
3. **Probability Formula:**
\[
\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} = \frac{5}{n}
\]
4. **Given Probability:**
\[
\frac{5}{n} = \frac{1}{10}
\]
5. **Solve for \( n \):**
\[
5 \times 10 = n \times 1 \\
n = 50
\]
**Therefore, there are \( \boxed{50} \) cards in the bag.**
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Bonus Knowledge
To find the number of cards in the bag, we know that the probability of drawing a card with a number less than or equal to 5 is given as \( \frac{1}{10} \). The number of favorable outcomes (drawing a card with a number less than or equal to 5) is 5 (if \( n \geq 5 \)). Thus, the probability can be expressed as: \[ P(\text{Number} \leq 5) = \frac{5}{n} \] Setting this probability equal to \( \frac{1}{10} \): \[ \frac{5}{n} = \frac{1}{10} \] Cross multiplying gives: \[ 5 \cdot 10 = 1 \cdot n \] So, \[ n = 50 \] Thus, there are \( 50 \) cards in the bag.
