Question

there are 1 red marble and 3 green marbles in a bag. if one marble is drawn, and then another, what is the probability that the first marble is red and the second marble is green? the probability is higher if the first marble drawn is dropdown before the second marble is drawn. the probability of drawing a red marble and then a green marble is dropdown if the first marble is replaced. the probability of drawing a red marble and then a green marble is dropdown if the first marble is not replaced.

Explanation:

Step1: Analyze replacement vs non - replacement

First, let's consider the total number of marbles initially. There are \(1 + 3=4\) marbles.

Case 1: With replacement

• Probability of drawing a red marble first: The number of red marbles is 1, so \(P(\text{red})=\frac{1}{4}\). Since we replace the marble, the total number of marbles remains 4 for the second draw. The number of green marbles is 3, so \(P(\text{green after red (replaced)})=\frac{1}{4}\times\frac{3}{4}=\frac{3}{16}\).

Case 2: Without replacement

• Probability of drawing a red marble first: \(P(\text{red})=\frac{1}{4}\). After drawing a red marble (and not replacing it), the total number of marbles becomes \(4 - 1 = 3\), and the number of green marbles is still 3. So \(P(\text{green after red (not replaced)})=\frac{1}{4}\times\frac{3}{3}=\frac{1}{4}\).
• Now, compare the two probabilities. \(\frac{1}{4}=\frac{4}{16}\) and \(\frac{3}{16}\). Since \(\frac{4}{16}>\frac{3}{16}\), the probability is higher when the first marble is not replaced.

Step2: Calculate probabilities

• For the first blank (the probability is higher if the first marble drawn is... before the second marble is drawn): We saw that without replacement (i.e., the first marble is not replaced) the probability is higher. So the first blank should be "not replaced".
• For the second blank (probability with replacement): As calculated, \(\frac{1}{4}\times\frac{3}{4}=\frac{3}{16}\).
• For the third blank (probability without replacement): \(\frac{1}{4}\times\frac{3}{3}=\frac{1}{4}\).

Answer:

• The probability is higher if the first marble drawn is \(\boldsymbol{\text{not replaced}}\) before the second marble is drawn.
• The probability of drawing a red marble and then a green marble is \(\boldsymbol{\frac{3}{16}}\) if the first marble is replaced.
• The probability of drawing a red marble and then a green marble is \(\boldsymbol{\frac{1}{4}}\) if the first marble is not replaced.