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A bag contains 6 red and 4 green balls. A fair dice is rolled and a number of balls equal to that appearing on the dice is chosen from the urn at random. The probability that all the balls selected are red is.
- a)1/3
- b)3/10
- c)1/8
- d)none
Correct answer is option 'D'. Can you explain this answer?
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Problem:
A bag contains 6 red and 4 green balls. A fair dice is rolled and a number of balls equal to that appearing on the dice is chosen from the urn at random. The probability that all the balls selected are red is.
Solution:
To find the probability of selecting all red balls, we need to consider two scenarios:
1. The dice shows a number less than or equal to 6.
2. The dice shows a number greater than 6.
Scenario 1: Dice shows a number less than or equal to 6
When the dice shows a number less than or equal to 6, the probability of selecting all red balls can be calculated by considering all possible outcomes of the dice and multiplying the probabilities of each outcome.
Outcomes:
- If the dice shows 1, there is only 1 ball to be selected. The probability of selecting a red ball is 6/10.
- If the dice shows 2, there are 2 balls to be selected. The probability of selecting 2 red balls is (6/10) * (5/9).
- If the dice shows 3, there are 3 balls to be selected. The probability of selecting 3 red balls is (6/10) * (5/9) * (4/8).
- If the dice shows 4, there are 4 balls to be selected. The probability of selecting 4 red balls is (6/10) * (5/9) * (4/8) * (3/7).
- If the dice shows 5, there are 5 balls to be selected. The probability of selecting 5 red balls is (6/10) * (5/9) * (4/8) * (3/7) * (2/6).
- If the dice shows 6, there are 6 balls to be selected. The probability of selecting 6 red balls is (6/10) * (5/9) * (4/8) * (3/7) * (2/6) * (1/5).
Calculating the probabilities:
To calculate the probability of each outcome, we multiply the probabilities of selecting red balls at each step. We can simplify the calculations as follows:
P(1) = (6/10) = 3/5
P(2) = (6/10) * (5/9) = 1/3
P(3) = (6/10) * (5/9) * (4/8) = 1/6
P(4) = (6/10) * (5/9) * (4/8) * (3/7) = 3/35
P(5) = (6/10) * (5/9) * (4/8) * (3/7) * (2/6) = 1/35
P(6) = (6/10) * (5/9) * (4/8) * (3/7) * (2/6) * (1/5) = 1/210
Scenario 2: Dice shows a number greater than 6
When the dice shows a number greater than 6, it is
A bag contains 6 red and 4 green balls. A fair dice is rolled and a number of balls equal to that appearing on the dice is chosen from the urn at random. The probability that all the balls selected are red is.
Solution:
To find the probability of selecting all red balls, we need to consider two scenarios:
1. The dice shows a number less than or equal to 6.
2. The dice shows a number greater than 6.
Scenario 1: Dice shows a number less than or equal to 6
When the dice shows a number less than or equal to 6, the probability of selecting all red balls can be calculated by considering all possible outcomes of the dice and multiplying the probabilities of each outcome.
Outcomes:
- If the dice shows 1, there is only 1 ball to be selected. The probability of selecting a red ball is 6/10.
- If the dice shows 2, there are 2 balls to be selected. The probability of selecting 2 red balls is (6/10) * (5/9).
- If the dice shows 3, there are 3 balls to be selected. The probability of selecting 3 red balls is (6/10) * (5/9) * (4/8).
- If the dice shows 4, there are 4 balls to be selected. The probability of selecting 4 red balls is (6/10) * (5/9) * (4/8) * (3/7).
- If the dice shows 5, there are 5 balls to be selected. The probability of selecting 5 red balls is (6/10) * (5/9) * (4/8) * (3/7) * (2/6).
- If the dice shows 6, there are 6 balls to be selected. The probability of selecting 6 red balls is (6/10) * (5/9) * (4/8) * (3/7) * (2/6) * (1/5).
Calculating the probabilities:
To calculate the probability of each outcome, we multiply the probabilities of selecting red balls at each step. We can simplify the calculations as follows:
P(1) = (6/10) = 3/5
P(2) = (6/10) * (5/9) = 1/3
P(3) = (6/10) * (5/9) * (4/8) = 1/6
P(4) = (6/10) * (5/9) * (4/8) * (3/7) = 3/35
P(5) = (6/10) * (5/9) * (4/8) * (3/7) * (2/6) = 1/35
P(6) = (6/10) * (5/9) * (4/8) * (3/7) * (2/6) * (1/5) = 1/210
Scenario 2: Dice shows a number greater than 6
When the dice shows a number greater than 6, it is
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A bag contains 6 red and 4 green balls. A fair dice is rolled and a number of balls equal to that appearing on the dice is chosen from the urnat random. The probability that all the balls selected are red is.a)1/3b)3/10c)1/8d)noneCorrect answer is option 'D'. Can you explain this answer?
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