Question

A bag contains 4 red, 5 blue and 3 green balls. One ball is drawn at random. The probability of getting a blue ball is:

• A. $\frac{1}{4}$
• B. $\frac{1}{3}$
• C. $\frac{5}{12}$
• D. $\frac{1}{2}$

Hint:

Always calculate the total sample pool size first. Once you know the denominator is 12, check if your target item count (5) can be simplified with it. Since 5 is a prime number, the fraction must remain exactly as $\frac{5}{12}$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Theoretical probability is the ratio of the number of favorable outcomes to the total number of possible outcomes in an experiment where all outcomes are equally likely.
In this problem, the "experiment" is drawing one ball. The "possible outcomes" consist of every individual ball inside the bag.
The "favorable outcome" refers to drawing a ball of the specific color requested, which is blue.
Step 2: Key Formula or Approach:
\[ \text{Probability } P(E) = \frac{n(E)}{n(S)} \]
Where:
\(n(E)\) = Number of favorable outcomes (number of blue balls)
\(n(S)\) = Total number of outcomes in the sample space (total balls)
Step 3: Detailed Explanation:
Let's determine the values for the components of our fraction.
First, calculate the total number of balls present in the bag (\(n(S)\)):
- Number of Red balls = \(4\)
- Number of Blue balls = \(5\)
- Number of Green balls = \(3\)
- Total number of balls = \(4 + 5 + 3 = 12\)

Next, identify the number of favorable outcomes (\(n(E)\)). Since we are asked for the probability of picking a blue ball, we look at the count of blue balls:
- Number of favorable outcomes = \(5\)

Substitute these values into the probability formula:
\[ P(\text{Blue}) = \frac{5}{12} \]
This fraction represents the likelihood of picking a blue ball. Can this be simplified further?
The numerator \(5\) is a prime number and is not a factor of the denominator \(12\). Thus, the fraction is in its simplest form.
Matching with the options provided:
(a) \(1/4\) is \(3/12\) (Incorrect)
(b) \(1/3\) is \(4/12\) (Incorrect)
(c) \(5/12\) (Correct)
(d) \(1/2\) is \(6/12\) (Incorrect)
Step 4: Final Answer:
The probability of getting a blue ball is \(\frac{5}{12}\).