There are two boxes. The red box contains four red marbles and one blue marble, and the blue box contains five blue marbles and three red marbles. The following experiment is done a hundred times: A marble is randomly drawn from one of the boxes, its color recorded, then replaced in the same box. The first drawing is from the red box, but each subsequent
drawing is determined by the color of the marble most recently drawn: if it is red, the next drawing comes from the red box; if it is blue, the next drawing comes from the blue box.
What is the probability that the first marble is red? the first marble is blue? the second marble is red? the second marble is blue? the hundredth marble is red?
the hundredth marble is blue?
The probability that the first marble is red is 4/9.
The probability that the first marble is blue is 5/9.
The probability that the second marble is red is 4/5.
The probability that the second marble is blue is 1/5.
The probability that the hundredth marble is red is 4/9.
The probability that the hundredth marble is blue is 5/9.
To find the probability of different events in this experiment, we need to carefully analyze the information given.
Let's go step by step to determine the probabilities:
1. Probability that the first marble is red:
Since the first drawing is from the red box, and the red box contains four red marbles and one blue marble, the probability of drawing a red marble is 4/5 or 0.8. So, the probability that the first marble is red is 0.8.
2. Probability that the first marble is blue:
Following the same reasoning, we know that the probability of drawing a blue marble from the red box is 1/5 or 0.2. Therefore, the probability that the first marble is blue is 0.2.
3. Probability that the second marble is red:
The probability of drawing a red marble as the second marble depends on the color of the first marble. If the first marble is red (which occurs with a probability of 0.8), we draw from the red box again, where the probability of drawing a red marble is 4/5. So the overall probability that the second marble is red is (0.8) * (4/5) = 0.64.
4. Probability that the second marble is blue:
Following the same logic, if the first marble is red (which occurs with a probability of 0.8), we draw from the red box again, where the probability of drawing a blue marble is 1/5. So the overall probability that the second marble is blue is (0.8) * (1/5) = 0.16.
5. Probability that the hundredth marble is red:
Given that the experiment follows the rule of drawing a marble from the same box as the previously drawn one, regardless of the color, the probability that the hundredth marble is red is the same as the probability that any marble drawn from the red box is red, which is 4/5 or 0.8.
In summary:
- Probability that the first marble is red: 0.8
- Probability that the first marble is blue: 0.2
- Probability that the second marble is red: 0.64
- Probability that the second marble is blue: 0.16
- Probability that the hundredth marble is red: 0.8
To find the probability of each event, we can analyze the situation step by step:
1. Probability that the first marble is red:
Since the first drawing is from the red box, which contains four red marbles and one blue marble, the probability of drawing a red marble is 4/5. Therefore, the probability of the first marble being red is 4/5.
2. Probability that the first marble is blue:
Similarly, the probability of drawing a blue marble from the red box is 1/5. Therefore, the probability of the first marble being blue is 1/5.
3. Probability that the second marble is red:
After drawing the first marble, the color of the second marble depends on the color of the first marble. If the first marble is red, the second drawing will be made from the red box (which still contains four red marbles and one blue marble). So, the probability of drawing a red marble as the second marble, given that the first marble was red, is 4/5.
4. Probability that the second marble is blue:
If the first marble is red, the probability of drawing a blue marble as the second marble, given that the first marble was red, is 1/5. This is because there is one blue marble in the red box.
5. Probability that the hundredth marble is red:
The same logic can be applied to the probability of the hundredth marble being red. At each step, the probability of drawing a red marble is 4/5, regardless of the number of previous steps.
Therefore, since each marble drawn is independent of the previous draws (as the marble is replaced back into the same box), the probability of the hundredth marble being red is also 4/5.