Question

determining the probability of independent events
regina has a bag of marbles that contains 3 blue marbles, 4 red marbles, and 5 yellow marbles. she draws one marble, then replaces it, and draws one more.
there are dropdown possible outcomes.
the probability the first drawn marble will be blue is dropdown.
the probability the second marble drawn will be yellow is dropdown.
so, p(blue, then yellow) = dropdown with options 1/15, 5/48, 1/4, 7/16

Explanation:

Step1: Calculate total number of marbles

First, we find the total number of marbles. There are 3 blue, 4 red, and 5 yellow marbles. So total marbles \( n = 3 + 4 + 5 = 12 \).

Step2: Probability the first marble is blue

The number of blue marbles is 3. The probability of drawing a blue marble first is the number of blue marbles divided by total marbles. So \( P(\text{blue first})=\frac{3}{12}=\frac{1}{4} \).

Step3: Probability the second marble is yellow (with replacement)

Since we replace the first marble, the total number of marbles remains 12, and the number of yellow marbles is 5. So the probability of drawing a yellow marble second is \( P(\text{yellow second})=\frac{5}{12} \).

Step4: Probability of blue then yellow

For independent events (since we replace the marble, the two draws are independent), the probability of both events happening is the product of their individual probabilities. So \( P(\text{blue then yellow}) = P(\text{blue first}) \times P(\text{yellow second})=\frac{1}{4} \times \frac{5}{12}=\frac{5}{48} \).

Answer:

• Number of possible outcomes: 12 (since total marbles are 12, and each draw is with replacement, but the number of possible outcomes for each draw is 12, and for two draws with replacement, it's \( 12\times12 = 144 \)? Wait, maybe the first part: "There are [ ] possible outcomes." Wait, the problem says she draws one marble, replaces it, and draws one more. So the number of possible outcomes for two draws with replacement: for each draw, 12 options, so total \( 12\times12 = 144 \). But maybe the first blank is about the number of possible outcomes for the two - draw experiment.

• Probability first marble is blue: \( \frac{1}{4} \)

• Probability second marble is yellow: \( \frac{5}{12} \)

• \( P(\text{blue, then yellow})=\frac{5}{48} \)

If we assume the blanks are:

1. Number of possible outcomes: \( 12\times12 = 144 \)

1. Probability first marble is blue: \( \frac{3}{3 + 4+5}=\frac{3}{12}=\frac{1}{4} \)

1. Probability second marble is yellow: \( \frac{5}{12} \)

1. \( P(\text{blue, then yellow})=\frac{1}{4}\times\frac{5}{12}=\frac{5}{48} \)

So filling the blanks:

• There are \(\boldsymbol{144}\) possible outcomes.

• The probability the first drawn marble will be blue is \(\boldsymbol{\frac{1}{4}}\).

• The probability the second marble drawn will be yellow is \(\boldsymbol{\frac{5}{12}}\).

• So, \( P(\text{blue, then yellow})=\boldsymbol{\frac{5}{48}} \)