Question

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 possible outcomes. the proba the first drawn marble will be blue is 1/4. the proba 3 the second marble drawn will be yellow is 5/12. so, p(blue x 4 yellow) = 5/48. 6 12

Explanation:

Step1: Calculate total number of marbles

Total marbles \(=3 + 4+5=12\)
Since she replaces the marble after first draw, for two - draw experiments (with replacement), the total number of possible outcomes is \(n = 12\times12=144\) (using the counting principle \(m\times n\), where \(m\) is the number of outcomes for the first event and \(n\) is the number of outcomes for the second event)

Step2: Verify probability of first - draw blue

Probability of an event \(P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). For the first draw, number of blue marbles \(= 3\), total marbles \(=12\). So \(P(\text{blue first})=\frac{3}{12}=\frac{1}{4}\)

Step3: Verify probability of second - draw yellow

For the second draw (after replacement), number of yellow marbles \(=5\), total marbles \(=12\). So \(P(\text{yellow second})=\frac{5}{12}\)

Step4: Calculate \(P(\text{blue first and yellow second})\)

Since the two draws are independent events (because of replacement), \(P(A\cap B)=P(A)\times P(B)\). Here \(A\) is the event of drawing a blue marble first and \(B\) is the event of drawing a yellow marble second. So \(P(\text{blue then yellow})=\frac{1}{4}\times\frac{5}{12}=\frac{5}{48}\)

Answer:

The number of possible outcomes is \(144\)