Probability of Drawing 3 Black Jelly Beans from a Mixed Bowl

Imagine a bowl filled with a variety of jelly beans. You're curious about the probability of drawing 3 black jelly beans in a row, one at a time, without replacing each jelly bean after it is drawn. This is a common problem in probability theory and can be approached in two main scenarios: drawing with and without replacement.

Understanding the Probability Theory

Let's delve into the steps involved in solving this problem and illustrate how the probabilities change when each jelly bean is either replaced or not.

Total Number of Jelly Beans

In the bowl, there are 12 red, 10 black, 5 white, and 3 green jelly beans. Therefore, the total number of jelly beans is:

Total number of jelly beans 12 10 5 3 30

Probability Calculation with Replacement

When drawing jelly beans with replacement, each draw is independent of the others. Therefore, the probability of drawing a black jelly bean remains the same for each draw.

Probability of drawing the first black jelly bean:

P1st black 10/30 1/3

Probability of drawing the second black jelly bean:

P2nd black 1/3

Probability of drawing the third black jelly bean:

P3rd black 1/3

Combined probability of drawing 3 black jelly beans with replacement:

P3 black (1/3) x (1/3) x (1/3) 1/27

Probability Calculation without Replacement

When drawing jelly beans without replacement, each draw affects the probability of the next draw. Let's walk through this step by step:

Probability of drawing the first black jelly bean:

P1st black 10/30 1/3

After drawing the first black jelly bean, there are now 9 black jelly beans left and 29 jelly beans in total:

P2nd black 1st black 9/29

After the second draw, 8 black jelly beans remain and 28 jelly beans are left:

P3rd black 1st and 2nd black 8/28 2/7

Combined probability of drawing 3 black jelly beans without replacement:

P3 black (1/3) x (9/29) x (2/7) 18/609

Conclusion

Depending on whether the jelly beans are replaced or not, the probability of drawing 3 black jelly beans changes significantly. When the jelly beans are not replaced, the probability is approximately 0.02956 or 18/609. In contrast, when the jelly beans are replaced, the probability is 1/27.

Understanding these concepts is essential for solving similar probability problems in a variety of real-world scenarios, such as in genetics, finance, and quality control. It's also a fundamental topic in statistics and probability theory.