Question

1. a bag of 50 marbles has 20 red, 15 blue, 10 green, and the rest are yellow marbles. if you select two marbles with replacement, what is the probability that you will select at least one red marble?

a \\(\frac{4}{5}\\)

b 0.4

c \\(\frac{16}{25}\\)

d 0.6

Explanation:

⚡ Using what you learned: independent and dependent events

Step 1: Identify the complement event

To find the probability of selecting "at least one red marble" in two selections with replacement, it is easier to calculate the probability of the complement event: selecting "no red marbles" in both draws.

$$ P(\text{at least one red}) = 1 - P(\text{no red on both draws}) $$

Step 2: Calculate the probability of selecting a non-red marble

The bag contains \(50\) marbles in total, and \(20\) of them are red. The number of non-red marbles is:

$$ 50 - 20 = 30 $$

Since the selection is done with replacement, the probability of selecting a non-red marble on any single draw is:

$$ P(\text{not red}) = \frac{30}{50} = \frac{3}{5} $$

Step 3: Calculate the probability of selecting no red marbles in two draws

Since the marbles are replaced, the two draws are independent events:

$$ P(\text{no red on both draws}) = P(\text{not red}) \times P(\text{not red}) = \frac{3}{5} \times \frac{3}{5} = \frac{9}{25} $$

Step 4: Subtract from 1 to find the final probability

$$ P(\text{at least one red}) = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} $$

Comparing this with the given options:

• A: \(\frac{4}{5}\)
• B: \(0.4\)
• C: \(\frac{16}{25}\)
• D: \(0.6\)

The correct option is C.

Answer:

C