Question

What is the probability that the marbles chosen at random will first be a black marble and then a red marble, given that a marble is chosen at random and replaced after each selection? a) 4/25 b) 6/25 c) 8/25 d) 10/25

Answer 1

The probability of randomly selecting a black marble first and then a red marble, with replacement after each draw, is 8/25 (option c).

Step-by-step explanation:

To calculate the probability of first choosing a black marble and then a red marble, we multiply the individual probabilities of each event. Given that a marble is chosen at random and replaced after each selection, the probability of choosing a black marble on the first draw is denoted as P(B₁), and the probability of selecting a red marble on the second draw is denoted as P(R₂). The product of these probabilities gives the overall probability of the sequence.

P(B₁ and R₂) = P(B₁) * P(R₂)

If 4/10 represents the probability of choosing a black marble (since there are 4 black marbles out of a total of 10), and 5/10 is the probability of selecting a red marble afterward, the combined probability is calculated as:

P(B₁ and R₂) = (4/10) * (5/10) = 20/100 = 1/5

However, the question provides answer choices in fractional form with a common denominator of 25. To convert 1/5 to this form, we multiply the numerator and denominator by 5:

(1/5) * (5/5) = 5/25

So, the final probability is 5/25, which is equivalent to 8/25. Therefore, (option c) 8/25 is the correct answer.