12 marbles are selected at random form a large collection of white , red, green and yellow marbles . The number of marbles of each color is unlimited. Find the probability that the selection contains at least one marble of each color ?
Created: 5 years ago |
Updated: 3 years ago
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12 marbles are selected at random form a large collection of white , red, green and yellow marbles . The number of marbles of each color is unlimited. Find the probability that the selection contains at least one marble of each color ?
This is same as W + R + G + Y = 12, where all are positive integers.
Step I..
first, give one to each, hence 4 gone. Now you can distribute the remaining 8 to anyone as the restriction of at least one is done.. W + R + G + Y = 8..W + R + G + Y = 8.. Now, take this as 8 marbles and 3 partitions, so total 11 things. You can choose these 3 partitions in 11C3 ways = 11!/8!3! = 11∗10∗9/3∗2 = 11∗5∗3 Step II..
Total ways..
W + R + G + Y = 12
Now, take this as 12 marbles and 3 partitions, so total 15 things. You can choose these 3 partitions in 15C3 ways = 15!/12!3! = 15∗14∗13/3∗2 = 5∗7∗13
Probability = 11∗5∗3/5∗7∗13 = 33/91
Step I..
first, give one to each, hence 4 gone. Now you can distribute the remaining 8 to anyone as the restriction of at least one is done.. W + R + G + Y = 8..W + R + G + Y = 8.. Now, take this as 8 marbles and 3 partitions, so total 11 things. You can choose these 3 partitions in 11C3 ways = 11!/8!3! = 11∗10∗9/3∗2 = 11∗5∗3 Step II..
Total ways..
W + R + G + Y = 12
Now, take this as 12 marbles and 3 partitions, so total 15 things. You can choose these 3 partitions in 15C3 ways = 15!/12!3! = 15∗14∗13/3∗2 = 5∗7∗13
Probability = 11∗5∗3/5∗7∗13 = 33/91
Md Azizur Rahman
Founder of Satt
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