Question

a bag with 8 marbles has 8 blue marbles. a marble is chosen from the bag at random. what is the probability that it is blue? write your answer as a fraction in simplest form.

Explanation:

Step1: Recall probability formula

The probability of an event is given by the number of favorable outcomes divided by the total number of possible outcomes. For this problem, the number of favorable outcomes (choosing a blue marble) is the number of blue marbles, and the total number of possible outcomes is the total number of marbles in the bag.
So the formula is \( P(\text{blue})=\frac{\text{Number of blue marbles}}{\text{Total number of marbles}} \).

Step2: Identify values

We know that the number of blue marbles is 8, and the total number of marbles in the bag is also 8.

Step3: Calculate probability

Substitute the values into the formula: \( P(\text{blue}) = \frac{8}{8} \).

Step4: Simplify the fraction

Simplify \( \frac{8}{8} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 8. So \( \frac{8\div8}{8\div8}=\frac{1}{1} = 1 \). Wait, actually, \( \frac{8}{8} \) simplifies to 1, but let's check again. Wait, the number of blue marbles is 8, total marbles is 8. So \( \frac{8}{8}=1 \), but let's confirm. Wait, probability of an event that is certain (since all marbles are blue) is 1. But let's do the steps properly.

Wait, step 1: Probability formula is \( P = \frac{\text{favorable}}{\text{total}} \). Step 2: Favorable (blue) = 8, total = 8. Step 3: \( P=\frac{8}{8} \). Step 4: Simplify \( \frac{8}{8} \) to 1 (since 8 divided by 8 is 1). But let's see, the problem says "write as a fraction in simplest form". \( \frac{8}{8} \) simplifies to \( \frac{1}{1} \) or just 1, but as a fraction, \( \frac{1}{1} \) or 1. Wait, but maybe I made a mistake. Wait, the bag has 8 marbles, all blue. So when we pick a marble, the probability it's blue is the number of blue marbles (8) over total marbles (8), which is \( \frac{8}{8}=1 \). So in simplest form, that's \( \frac{1}{1} \) or 1, but typically written as 1 (which is \( \frac{1}{1} \) but simplified). Wait, but let's check the steps again.

Wait, maybe I messed up the numbers. Wait, the problem says "a bag with 8 marbles has 8 blue marbles". So total marbles \( n = 8 \), number of blue marbles \( m = 8 \). So probability \( P = \frac{m}{n}=\frac{8}{8}=1 \). So in simplest form, that's 1, which is \( \frac{1}{1} \), but usually written as 1. But let's confirm the simplification. The greatest common divisor of 8 and 8 is 8, so divide numerator and denominator by 8: \( \frac{8\div8}{8\div8}=\frac{1}{1}=1 \). So the probability is 1.

Answer:

\( \frac{1}{1} \) (or simply 1, but as a fraction in simplest form, \( \frac{1}{1} \) is correct, though often written as 1)