Mrs. Dwayne has three boxes. The first box contains three red marbles and five blue marbles. The second box has four red marbles and six blue marbles. The third box has five red marbles and seven blue marbles. Mrs Dwayne pulls a marble from each box five times. but before her next turn, returns the marble to the box it came from. A red marble was pulled two time for box 1, four time from box 2, and three times for box 3.

Question

Mrs. Dwayne has three boxes. The first box contains three red marbles and five blue marbles. The second box has four red marbles and six blue marbles. The third box has five red marbles and seven blue marbles. Mrs Dwayne pulls a marble from each box five times. but before her next turn, returns the marble to the box it came from. A red marble was pulled two time for box 1, four time from box 2, and three times for box 3.

Determine the theoretical and experimental probability of pulling a red marble from each box. For which box was the experimental probability closet to the theoretical probability? Explain your answer by providing proof and justification.

Answer 1

a) Here's the theoretical probability, it's the probability that comes up from calculations using known formulae of probability:

P(red from all three) = P(red from A)*P(red from B)*P(red from C)
P(A) = 3/8 (3 out of 8)
P(B) = 4/10
P(C) = 5/12

= (3/8)*(4/10)*(5/12)
= 1/16

b) Experimental Probability of pulling out a red marble:
For A: 2/5 (Two times out of 5)
For B: 4/5
For C: 3/5
Experimental Prob. of pulling out a red marble from all three: 2/5*4/5*3/5 = 24/125

Try comparing the theoretical and experimental probabilities of the boxes on your own, and you'll find the answer

Answer 2

To determine the theoretical probability of pulling a red marble from each box, we need to calculate the total number of marbles and the number of red marbles in each box.

First box:
Total marbles = 3 red marbles + 5 blue marbles = 8 marbles
Number of red marbles = 3

Second box:
Total marbles = 4 red marbles + 6 blue marbles = 10 marbles
Number of red marbles = 4

Third box:
Total marbles = 5 red marbles + 7 blue marbles = 12 marbles
Number of red marbles = 5

Now, let's calculate the theoretical probability for each box:

Theoretical probability for the first box = Number of red marbles / Total number of marbles
= 3 / 8 = 3/8

Theoretical probability for the second box = 4 / 10 = 2/5

Theoretical probability for the third box = 5 / 12

Now, let's calculate the experimental probability based on the given data. We know that Mrs. Dwayne pulled a marble from each box five times.

Experimental probability for the first box = Number of times a red marble was pulled / Total number of times (5)
= 2 / 5 = 2/5

Experimental probability for the second box = 4 / 5 = 4/5

Experimental probability for the third box = 3 / 5 = 3/5

Comparing the experimental probabilities with the theoretical probabilities, we can see that the experimental probability for the second box is closest to its theoretical probability. The experimental probability for box 2 is 4/5, while the theoretical probability is 2/5. This shows that the results from the experiment align more closely with the predicted outcome for box 2.

To provide proof and justification, we can calculate the difference between the experimental and theoretical probabilities for each box:

Difference for box 1:
Experimental probability - Theoretical probability = (2/5) - (3/8) = 1/40

Difference for box 2:
Experimental probability - Theoretical probability = (4/5) - (2/5) = 2/5

Difference for box 3:
Experimental probability - Theoretical probability = (3/5) - (5/12) = 9/60

Based on the calculations above, we can see that the difference is smallest for box 2 (2/5). This indicates that the experimental probability is closest to the theoretical probability for this box. Therefore, the experimental probability of pulling a red marble from box 2 was closest to the theoretical probability.