Probability Calculation of Drawing Blue Marbles Without Replacement
Consider the scenario where a bag contains a specific number of marbles of different colors. The bag includes four red marbles, three blue marbles, two yellow marbles, and one green marble. In this article, we will calculate the probability of drawing two blue marbles consecutively without replacement. We will explore both joint and conditional probability calculations to solve this problem.
Understanding the Problem
The total number of marbles in the bag is 10 (4 red 3 blue 2 yellow 1 green).
Calculating the Probability of Drawing Two Blue Marbles
To find the probability of drawing two blue marbles consecutively without replacement, we need to consider the sequence of draws and the changing probabilities after each draw.
Joint Probability of Drawing Two Blue Marbles
The first step is to calculate the joint probability of drawing a blue marble on both the first and second draws.
Total ways two draws can be drawn: 10 x 9 90
Number of favorable outcomes for the first draw (blue): 3
Number of favorable outcomes for the second draw (blue, knowing the first was blue): 2
Total favorable outcomes: 3 x 2 6
Joint probability: 6/90 1/15
Conditional Probability of Drawing a Blue Marble on the Second Draw Given the First Draw Is Red
Let's calculate the conditional probability of drawing a blue marble on the second draw given the first draw is red.
Total ways to draw a red marble first: 4
After drawing one red marble, total marbles left: 9 (3 blue 2 yellow 1 green)
Number of favorable outcomes for drawing a blue marble second: 3
Probability of drawing a blue marble on the second draw given the first draw was red:
Probability number of favorable outcomes / number of possible outcomes 3 / 9 1 / 3
Therefore, the conditional probability Pr(2nd blue | 1st red) (1 / 15) / (1 / 3) 1 / 5
Answering the Given Questions
How many red marbles are there?
Answer: 4
How many marbles are there in total?
Answer: 10
What fraction is the first answer over the second answer?
Answer: 4/10 2/5
Reduce 2/5.
Answer: 2/5
Conclusion
Probability calculations, especially in scenarios involving non-replacement events, can be complex but are essential in understanding the likelihood of various outcomes. By breaking down the problem into smaller, manageable steps, we can accurately determine probabilities such as the likelihood of drawing blue marbles in sequence from a mixed bag of colored marbles.