To tackle this probability question, let's break it down step by step. We have an urn containing 6 white balls and 4 red balls. When we draw a ball, we note its color and then add another ball of the same color back into the urn. This process affects the total number of balls and the composition of colors in the urn, which we need to consider when calculating the probabilities.
Understanding the Initial Setup
Initially, the urn contains:
- 6 white balls
- 4 red balls
This gives us a total of 10 balls in the urn (6 + 4 = 10).
Calculating the Probabilities
When we draw a ball, we have two possible outcomes: drawing a red ball or drawing a white ball. Let's calculate the probabilities for each scenario.
Probability of Drawing a Red Ball
The probability of selecting a red ball on the first draw can be calculated using the formula:
P(Red) = Number of Red Balls / Total Number of Balls
Substituting the values:
P(Red) = 4 / 10 = 0.4
Probability of Drawing a White Ball
Similarly, the probability of selecting a white ball is:
P(White) = Number of White Balls / Total Number of Balls
Substituting the values:
P(White) = 6 / 10 = 0.6
Impact of Adding a Ball
After we draw a ball and note its color, we add another ball of the same color back into the urn. This changes the total number of balls and the composition:
- If we draw a red ball, the new composition will be 6 white balls and 5 red balls (total 11 balls).
- If we draw a white ball, the new composition will be 7 white balls and 4 red balls (total 11 balls).
Revisiting the Probabilities
Now, let's consider the probabilities again after adding the ball:
After Drawing a Red Ball
If we drew a red ball first, the new probabilities are:
P(Red after Red) = 5 / 11
P(White after Red) = 6 / 11
After Drawing a White Ball
If we drew a white ball first, the new probabilities are:
P(Red after White) = 4 / 11
P(White after White) = 7 / 11
Final Probability Calculation
To find the overall probabilities of selecting a red or white ball after the first draw, we can use the law of total probability:
P(Red) = P(Red first) * P(Red after Red) + P(White first) * P(Red after White)
Substituting the values:
P(Red) = (0.4 * 5/11) + (0.6 * 4/11)
P(Red) = (2 / 11) + (2.4 / 11) = 4.4 / 11 ≈ 0.4
For white, we do a similar calculation:
P(White) = P(Red first) * P(White after Red) + P(White first) * P(White after White)
P(White) = (0.4 * 6/11) + (0.6 * 7/11)
P(White) = (2.4 / 11) + (4.2 / 11) = 6.6 / 11 ≈ 0.6
Conclusion
In summary, the probabilities of selecting a red or white ball after the first draw and adding another ball of the same color are approximately:
- A) Probability of selecting 1 red ball: 0.4
- B) Probability of selecting 1 white ball: 0.6
This illustrates how the process of drawing and replacing a ball affects the overall probabilities in a dynamic way. If you have any further questions or need clarification on any part of this, feel free to ask!
