Red Marbles And 8 Blue Marbles Are Placed Into A Bag

1 2 Red Marbles And 8 Blue Marbles Are Placed Into A Bag Bob Mixes

1. 2 red marbles and 8 blue marbles are placed into a bag. Bob mixes up the bag and randomly selects a marble. He continues to do so, replacing the marble after each selection, until a red marble is selected. (12 points) a. What is the probability that the first time that a red marble is pulled is on Bob’s 8th try? b. On average, how many marbles will Bob have to pull in order to get a red marble? (Hint: use math expectation)

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In this problem, we analyze a scenario of repeated independent trials where the outcome of each trial depends on selecting a marble from a bag containing red and blue marbles with replacement. This setup closely relates to the concept of geometric distribution, which models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials.

Part (a) involves calculating the probability that the first successful red marble appears exactly on the 8th try. Since the marbles are replaced each time, the probability of selecting a red marble in any individual trial remains constant at 2/10 or 0.2, given that there are 2 red marbles out of 10 total marbles. Consequently, the probability that the first 7 trials result in blue marbles (failures) and the 8th trial results in a red marble (success) is computed as:

P(first success on 8th try) = (probability of failure)^7 × (probability of success) = (8/10)^7 × (2/10) = (0.8)^7 × 0.2 ≈ 0.2097 × 0.2 ≈ 0.0419.

Part (b) asks for the expected number of marbles Bob needs to pull to get a red marble. Recognizing this as the expected value of a geometric distribution with success probability p = 0.2, the expectation is given by:

E(X) = 1/p = 1/0.2 = 5.

This result implies that, on average, Bob will need to draw 5 marbles to obtain a red marble.

In conclusion, the probability calculations demonstrate the stochastic nature of sequential trials with replacement, and the expectation clarifies the typical number of draws for a success in such processes. These concepts are pertinent in fields such as quality control, reliability testing, and probabilistic modeling where similar Bernoulli processes occur.

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