Question

3. MARBLES A bag contains 4 red, 5 green, and 2 blue marbles. You randomly draw 2 marbles, one at a time. Find the probabililty that both are red if: a. you replace the first marble. b. you do not replace the first marble.

Answer 1

• a) 16/121 or 13.2%
• b) 6/55 or 10.9%

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Scenario a

The first marble is replaced.

In this case, after picking up the first red marble, we have replaced it.

Hence, the total number of marbles in the bag remains the same, which is:

• 4 (red) + 5 (green) + 2 (blue) = 11 marbles

The probability of drawing a red marble on the first draw is thus, the number of red marbles divided by the total:

• 4 out of 11

Then, since we are replacing the marble, for the second draw, the probability remains the same.

By multiplying the probabilities of the two independent events we can obtain the overall probability of both these events happening together.

• (4/11) × (4/11) = 16/121 or 13.2%

Scenario b

The first marble is not replaced

In this case, after picking up the first red marble, we have not replaced it.

Hence, the total number of marbles in the bag decreases, which is now:

• 4 (red, one less now) + 5 (green) + 2 (blue) = 10 marbles

The probability of drawing a red marble on the first draw is again 4 out of 11.

However, since we are not replacing the marble, for the second draw, the probability changes.

Now, we only have 3 red marbles, out of a total of 10.

Again, we need to multiply the probabilities of the two independent events to calculate the overall probability of both these events occurring.:

• 4/11 × 3/10 = 6/55 or 10.9%