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In a bag, there are 3 red balls, 4 blue balls, and 5 green balls. What is the probability of picking a red ball?

$\frac{1}{4}$

$\frac{3}{12}$

$\frac{1}{3}$

$\frac{5}{12}$

Correct Answer

⚙️ TE • Technical Direct FormulaMathematicsStatistics and Probability

Option A

$\frac{1}{4}$

Quick Summary:

Directly sum all the balls to find the total sample space, then write the ratio of red balls to the total and simplify the fraction immediately.

📐MAMath SolutionDirect Formula

📋 Given

Number of red balls = 3, Number of blue balls = 4, Number of green balls = 5

🔢 Formula Used

P(E)=Number of favorable outcomesTotal number of possible outcomesP(E) = \frac{\text{\text{Number} of favorable outcomes}}{\text{\text{Total} number of possible outcomes}}P(E)=Total number of possible outcomesNumber of favorable outcomes​

⚡ Exam Hall Shortcut / Speed Trick

Directly sum all the balls to find the total sample space, then write the ratio of red balls to the total and simplify the fraction immediately.

⚠️ Common Student Trap / Pitfall

Students often forget to sum all the balls together to find the total number of outcomes, or fail to simplify the fraction to its lowest terms.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Find Total Number of Balls

Add the number of red, blue, and green balls to find the total number of outcomes in the sample space: $3 + 4 + 5 = 12$ .

extTotaloutcomes=3+4+5=12ext{\text{Total} outcomes} = 3 + 4 + 5 = 12extTotaloutcomes=3+4+5=12

Identify Favorable Outcomes

The number of red balls represents the favorable outcomes, which is given as $3$ .

extFavorableoutcomes=3ext{\text{Favorable} outcomes} = 3extFavorableoutcomes=3

Apply Probability Formula

Substitute the values into the probability formula: $P(\text{red}) = \frac{3}{12}$ .

P(red)=312P(\text{red}) = \frac{3}{12}P(red)=123​

Simplify the Fraction

Reduce the fraction $\frac{3}{12}$ to its simplest form by dividing the numerator and denominator by $3$ , yielding $\frac{1}{4}$ .

rac312=14rac{3}{12} = \frac{1}{4}rac312=41​

✅

A is correct because the probability of picking a red ball is calculated as $\frac{3}{12}$ , which simplifies to $\frac{1}{4}$ .

Core Concepts Used

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Probability Sample Space Favorable Outcomes Simplification of Fractions

💡 EXAM TIP

Probability concepts frequently overlap with Permutations and Combinations in competitive exams when selecting multiple items from a bag.

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