If a bag contains 8 red marbles and 4 blue marbles, what is the probability of selecting 3 marbles successively with replacement and all 3 marbles being blue?
To calculate the probability of selecting 3 marbles successively with replacement and all 3 marbles being blue, we need to follow the following steps:
Step 1: Determine the total number of marbles in the bag.
The bag contains 8 red marbles and 4 blue marbles, so the total number of marbles in the bag is 8 + 4 = 12.
Step 2: Determine the probability of selecting a blue marble.
Since there are 4 blue marbles in the bag out of a total of 12 marbles, the probability of selecting a blue marble is 4/12 = 1/3.
Step 3: Determine the probability of selecting 3 blue marbles in succession.
Since we are selecting with replacement, the probability of selecting a blue marble on each draw remains the same. So, we can use the multiplication rule.
The probability of selecting a blue marble on the first draw is 1/3.
The probability of selecting a blue marble on the second draw is also 1/3.
The probability of selecting a blue marble on the third draw is also 1/3.
To find the probability of all three events happening, we multiply the individual probabilities together:
P(all 3 marbles are blue) = (1/3) * (1/3) * (1/3)
= 1/27
Therefore, the probability of selecting 3 marbles successively with replacement and all 3 marbles being blue is 1/27.
To find the probability of selecting 3 marbles successively with replacement and all 3 marbles being blue, we can break down the problem step-by-step.
Step 1: Determine the probability of selecting a blue marble on each draw.
Since there are 4 blue marbles and a total of 12 marbles (8 red + 4 blue), the probability of selecting a blue marble on each draw is 4/12 or 1/3.
Step 2: Calculate the probability of selecting a blue marble on the first draw, second draw, and third draw.
Since each draw is done with replacement, the probability of selecting a blue marble on the first, second, and third draw is the same and can be calculated using the probability from Step 1.
Probability of a blue marble on the first draw: 1/3
Probability of a blue marble on the second draw: 1/3
Probability of a blue marble on the third draw: 1/3
Step 3: Multiply the probabilities of each draw.
Since each draw is independent, we can multiply the probabilities of each draw to find the probability of all three marbles being blue.
Probability of all three marbles being blue = (1/3) * (1/3) * (1/3) = 1/27
Therefore, the probability of selecting 3 marbles successively with replacement and all 3 marbles being blue is 1/27.