Understanding the Probability of Drawing Marbles with Combinations

Conducting probability calculations, especially when dealing with specific combinations of marbles, can sometimes be a challenging yet fascinating topic. This article aims to break down the process of finding the probability of drawing exactly 1 blue and 2 red marbles from a bag containing 6 red and 4 blue marbles. By using the power of combinations, we can determine the exact probability of this event happening. For a deeper dive into such calculations and more probability problems, check out my Quora Profile.

Step-by-Step Calculation

To understand the calculation of the probability of drawing exactly 1 blue and 2 red marbles from a bag containing 6 red and 4 blue marbles, let's go through the problem-solving process step-by-step.

Determining the Total Number of Ways to Draw 3 Marbles

The first step is to determine the total number of ways to draw 3 marbles from a total of 10 marbles. Here’s how you can calculate it using the combination formula:

Step 1: Calculate the total combinations of drawing 3 marbles from 10.

The formula for combinations is:

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

Where (n) is the total number of items, and (k) is the number of items to choose.

Total combinations:

(binom{10}{3} frac{10!}{3!(10-3)!} frac{10 times 9 times 8}{3 times 2 times 1} 120)

Calculating the Number of Favorable Outcomes

Next, we need to determine the number of ways to draw 1 blue marble and 2 red marbles.

Step 2: Choosing 1 Blue Marble from 4

(binom{4}{1} 4)

Step 3: Choosing 2 Red Marbles from 6

(binom{6}{2} frac{6!}{2!(6-2)!} frac{6 times 5}{2 times 1} 15)

Calculating the Total Combinations for 1 Blue and 2 Red Marbles

Now, multiply the number of ways to choose the blue and red marbles:

Total combinations for 1 blue and 2 red:

(text{Total combinations} binom{4}{1} times binom{6}{2} 4 times 15 60)

Calculating the Probability

The probability (P) of drawing exactly 1 blue marble and 2 red marbles is given by the ratio of the favorable outcomes to the total outcomes:

(P(text{1 blue, 2 red}) frac{text{Number of favorable outcomes}}{text{Total outcomes}} frac{60}{120} frac{1}{2})

Conclusion

Therefore, the probability of drawing exactly 1 blue marble and 2 red marbles is (frac{1}{2}).

For a deeper understanding and more probability problems, visit my Quora Profile to see detailed solutions and insights on various probability questions.