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A bag contains 8 balls, of which 4 are white and the remaining are red. Three balls are selected from the bag at random without replacement. Find the probability that at least one of the three selected balls is red.
- a)1/14
- b)5/14
- c)9/14
- d)11/14
- e)13/14
Correct answer is option 'E'. Can you explain this answer?
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Verified Answer
A bag contains 8 balls, of which 4 are white and the remaining are red...
Step I: Define Non-event
We are given that from a bag containing 4 red and 4 white balls, three balls are selected at random without replacement. We have to find the probability that at least one red ball is selected.
Now, following are the events in this case:
Event 1: One red ball is selected
Event 2: Two red balls are selected
Event 3: Three red balls are selected
However, there is only one non-event i.e. no red ball is selected. So, it will be easier to solve this question using the non-event approach.
So, we can write the probability event equation as:
P (At least one red ball is selected) = 1 – P (No red ball is selected)
Step II: Find n, the number of ways in which all outcomes can occur
Now, the total number of ways in which 3 balls can be selected from 8 balls is:
= 8C3 = 56
So, n = 8C3 = 56
Step III: Find x, the number of ways in which the Non-event can occur
Next, we are going to find the number of ways in which the non-event can occur.
As mentioned above, the non-event is that no red ball is selected among the three balls. This implies that the three selected balls are all white.
Now, the total number of white balls is 4.
So, we have to find the number of ways in which 3 white balls can be selected from 4 white balls.
Thus, x = 4C3 = 4
Step IV: Calculate probability for Non-event
P(No red ball is selected)
Step V: Probability (Event) = 1-Probability (Non-event)
P (At least one red ball is selected) = 1 – P (No red ball is selected)
Answer: Option (E)
Most Upvoted Answer
A bag contains 8 balls, of which 4 are white and the remaining are red...
To find the probability that at least one of the three selected balls is red, we can use the concept of complementary probability.
Complementary Probability: The probability of an event not occurring is 1 minus the probability of the event occurring.
Let's break down the problem into steps:
Step 1: Calculate the probability of selecting three white balls.
The probability of selecting the first white ball is 4/8 since there are 4 white balls out of 8 total balls.
After selecting the first white ball, the probability of selecting the second white ball is 3/7 since there are 3 white balls remaining out of 7 total balls.
Similarly, the probability of selecting the third white ball is 2/6 since there are 2 white balls remaining out of 6 total balls.
Therefore, the probability of selecting three white balls is (4/8) * (3/7) * (2/6) = 1/14.
Step 2: Calculate the probability of selecting at least one red ball.
As mentioned earlier, the complementary probability is used here.
The probability of selecting at least one red ball is 1 - (probability of selecting three white balls).
Therefore, the probability of selecting at least one red ball is 1 - 1/14 = 13/14.
Thus, the correct answer is option 'E' (13/14).
Complementary Probability: The probability of an event not occurring is 1 minus the probability of the event occurring.
Let's break down the problem into steps:
Step 1: Calculate the probability of selecting three white balls.
The probability of selecting the first white ball is 4/8 since there are 4 white balls out of 8 total balls.
After selecting the first white ball, the probability of selecting the second white ball is 3/7 since there are 3 white balls remaining out of 7 total balls.
Similarly, the probability of selecting the third white ball is 2/6 since there are 2 white balls remaining out of 6 total balls.
Therefore, the probability of selecting three white balls is (4/8) * (3/7) * (2/6) = 1/14.
Step 2: Calculate the probability of selecting at least one red ball.
As mentioned earlier, the complementary probability is used here.
The probability of selecting at least one red ball is 1 - (probability of selecting three white balls).
Therefore, the probability of selecting at least one red ball is 1 - 1/14 = 13/14.
Thus, the correct answer is option 'E' (13/14).
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