Questions: A bag contains n cards, each having one of the numbers (1,2,3, ..., 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 1/2. How many cards are in the bag?

A bag contains n cards, each having one of the numbers (1,2,3, ..., 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 1/2. How many cards are in the bag?

Transcript text: 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 $\frac{1}{2}$. How many cards are in the bag?

Solution

Solution Steps

To find the number of cards in the bag, we need to determine the value of $ n $ such that the probability of drawing a card with a number less than or equal to 5 is $\frac{1}{2}$. This means that the number of favorable outcomes (cards numbered 1 to 5) is half of the total number of outcomes (total cards $ n $). Therefore, we set up the equation $\frac{5}{n} = \frac{1}{2}$ and solve for $ n $.

Step 1: Set Up the Probability Equation

The problem states that the probability of drawing a card with a number less than or equal to 5 is $\frac{1}{2}$. This can be expressed as: \[ \frac{5}{n} = \frac{1}{2} \] where $ n $ is the total number of cards in the bag.

Step 2: Solve for $ n $

To find $ n $, we solve the equation: \[ 5 = \frac{n}{2} \] Multiplying both sides by 2 gives: \[ n = 10 \]