Probability of Drawing Same Color Marbles from a Bag

Probability is a fundamental concept in mathematics, often applied in various fields such as statistics, finance, and engineering. One classic problem in probability involves determining the likelihood of specific events occurring when drawing marbles from a bag. Let's explore the probability of drawing two marbles of the same color from a bag containing 3 red, 4 yellow, and 5 blue marbles.

Understanding the Problem

The problem statement involves three different colored marbles: red, yellow, and blue. The bag has the following distribution: 3 red marbles 4 yellow marbles 5 blue marbles We need to determine the probability of drawing two marbles of the same color from this bag without replacement.

Total Number of Marbles and Combinations

First, we calculate the total number of marbles in the bag:

Total number of marbles 3 (red) 4 (yellow) 5 (blue) 12 marbles

The total number of ways to choose 2 marbles from 12 can be calculated using combinations (denoted as ( binom{n}{r} )), where ( n ) is the total number of items, and ( r ) is the number of items to choose. The formula for combinations is:

( binom{n}{r} frac{n!}{r!(n-r)!} )

Applying the formula to our situation:

( binom{12}{2} frac{12!}{2!(12-2)!} frac{12 times 11}{2 times 1} 66 ) ways

Ways to Choose Marbles of the Same Color

Next, we need to determine the number of ways to draw 2 marbles of the same color for each color: Red Marbles: ( binom{3}{2} frac{3!}{2!(3-2)!} frac{3 times 2}{2 times 1} 3 ) Yellow Marbles: ( binom{4}{2} frac{4!}{2!(4-2)!} frac{4 times 3}{2 times 1} 6 ) Blue Marbles: ( binom{5}{2} frac{5!}{2!(5-2)!} frac{5 times 4}{2 times 1} 10 ) The total number of ways to draw 2 marbles of the same color is the sum of these combinations:

Total same color 3 (red) 6 (yellow) 10 (blue) 19

Calculating the Probability

The probability that both marbles drawn are of the same color is given by the ratio of the number of favorable outcomes to the total outcomes:

P(same color) ( frac{19}{66} )

This fraction represents the probability that both drawn marbles will be of the same color.

Further Explorations

When the selections occur without replacement, the problem can be re-evaluated using a different approach. Here is one of the solutions: Possible outcomes: ( 345^2 144 ) Favorable outcomes for A: [3452 one R][33 two R]54963 Required probability for A: ( frac{63}{144} frac{7}{16} 0.4375 ) Favorable outcomes for B: 64 Required probability for B: ( frac{64}{144} frac{4}{9} 0.4444... ) For another possible scenario with different outcomes, the probabilities are adjusted as follows: Possible outcomes: 345345–11211132 Favorable outcomes for A: 60 Probability for A: ( frac{60}{132} frac{5}{11} 0.4545... ) Favorable outcomes for B: 64 Probability for B: ( frac{64}{132} frac{16}{33} 0.4848... )

Conclusion

In conclusion, the probability of drawing two marbles of the same color from a bag containing 3 red, 4 yellow, and 5 blue marbles, when the selections occur without replacement, is approximately 0.4444. Understanding such problems helps in developing problem-solving skills and applying probability concepts effectively in various real-world scenarios.