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Example 13 - A committee of two persons is selected from two - Examples

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  1. Chapter 16 Class 11 Probability
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Example 13 A committee of two persons is selected from two men and two women. What is the probability that the committee will have (a) no man? If no man is selected, it means only women are selected So, we have to select 2 women Total number of persons = 2 + 2 = 4 Number of persons to be selected = 2 P(no man is selected) = ๏ท๐‘ต๐’–๐’Ž๐’ƒ๐’†๐’“ ๐’๐’‡ ๐’˜๐’‚๐’š๐’” ๐’ƒ๐’๐’•๐’‰ ๐’˜๐’๐’Ž๐’†๐’ ๐’Š๐’” ๐’”๐’†๐’๐’†๐’„๐’•๐’†๐’…๏ทฎ๐‘ป๐’๐’•๐’‚๐’ ๐’๐’–๐’Ž๐’ƒ๐’†๐’“ ๐’๐’‡ ๐’˜๐’‚๐’š๐’”๏ทฏ Total Number of ways = 4C2 = ๏ท4!๏ทฎ2!๏ท4 โˆ’2๏ทฏ!๏ทฏ = ๏ท4!๏ทฎ2!๏ท2๏ทฏ!๏ทฏ = ๏ท4 ร— 3 ร— 2!๏ทฎ2 ร— 1 ร— 2!๏ทฏ = 6 Number of women = 2 Number of women to be selected = 2 Number of ways both women is selected = 2C2 = ๏ท2!๏ทฎ2!๏ท2 โˆ’2๏ทฏ!๏ทฏ = ๏ท2!๏ทฎ2!0!๏ทฏ = ๏ท2!๏ทฎ2! ร— 1๏ทฏ = 1 ways Putting values in (1) P(no man is selected) = ๏ท๐‘๐‘ข๐‘š๐‘๐‘’๐‘Ÿ ๐‘œ๐‘“ ๐‘ค๐‘Ž๐‘ฆ๐‘  ๐‘๐‘œ๐‘กโ„Ž ๐‘ค๐‘œ๐‘š๐‘’๐‘› ๐‘–๐‘  ๐‘ ๐‘’๐‘™๐‘’๐‘๐‘ก๐‘’๐‘‘๏ทฎ๐‘‡๐‘œ๐‘ก๐‘Ž๐‘™ ๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ ๐‘œ๐‘“ ๐‘ค๐‘Ž๐‘ฆ๐‘ ๏ทฏ = ๏ท๐Ÿ๏ทฎ๐Ÿ”๏ทฏ Example 13 A committee of two persons is selected from two men and two women. What is the probability that the committee will have (b) one man ? Committee will have one man means one man & one woman is selected P(one man is selected) = ๏ท๐‘ต๐’–๐’Ž๐’ƒ๐’†๐’“ ๐’๐’‡ ๐’˜๐’‚๐’š๐’” ๐Ÿ ๐’Ž๐’‚๐’ ๐’Š๐’” ๐’”๐’†๐’๐’†๐’„๐’•๐’†๐’… ร— ๐‘ต๐’–๐’Ž๐’ƒ๐’†๐’“ ๐’๐’‡ ๐’˜๐’‚๐’š๐’” ๐Ÿ ๐’˜๐’๐’Ž๐’‚๐’ ๐’Š๐’” ๐’”๐’†๐’๐’†๐’„๐’•๐’†๐’… ๏ทฎ๐‘ป๐’๐’•๐’‚๐’ ๐’๐’–๐’Ž๐’ƒ๐’†๐’“ ๐’๐’‡ ๐’˜๐’‚๐’š๐’”๏ทฏ Total Number of men = 2 Number of men to be selected = 1 Number of ways to select 1 man = 2C1 = ๏ท2!๏ทฎ1!๏ท2 โˆ’1๏ทฏ!๏ทฏ = ๏ท2!๏ทฎ1!๏ทฏ = 2 ways Total Number of women = 2 Number of women to be selected = 1 Number of ways to select 1 woman = 2C1 = ๏ท2!๏ทฎ1!๏ท2 โˆ’1๏ทฏ!๏ทฏ = ๏ท2!๏ทฎ1!๏ทฏ = 2 ways P(one man is selected) = ๏ท๐‘๐‘ข๐‘š๐‘๐‘’๐‘Ÿ ๐‘œ๐‘“ ๐‘ค๐‘Ž๐‘ฆ๐‘  1 ๐‘š๐‘Ž๐‘› ๐‘–๐‘  ๐‘ ๐‘’๐‘™๐‘’๐‘๐‘ก๐‘’๐‘‘ ร— ๐‘๐‘ข๐‘š๐‘๐‘’๐‘Ÿ ๐‘œ๐‘“ ๐‘ค๐‘Ž๐‘ฆ๐‘  1 ๐‘ค๐‘œ๐‘š๐‘Ž๐‘› ๐‘–๐‘  ๐‘ ๐‘’๐‘™๐‘’๐‘๐‘ก๐‘’๐‘‘ ๏ทฎ๐‘‡๐‘œ๐‘ก๐‘Ž๐‘™ ๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ ๐‘œ๐‘“ ๐‘ค๐‘Ž๐‘ฆ๐‘ ๏ทฏ = ๏ท2 ร— 2๏ทฎ6๏ทฏ = ๏ท4๏ทฎ6๏ทฏ = ๏ท๐Ÿ๏ทฎ๐Ÿ‘๏ทฏ Example 13 A committee of two persons is selected from two men and two women. What is the probability that the committee will have (c) two man ? P(two men are selected) = ๏ท๐‘ต๐’–๐’Ž๐’ƒ๐’†๐’“ ๐’๐’‡ ๐’˜๐’‚๐’š๐’” ๐’ƒ๐’๐’•๐’‰ ๐’Ž๐’†๐’ ๐’Š๐’” ๐’”๐’†๐’๐’†๐’„๐’•๐’†๐’…๏ทฎ๐‘ป๐’๐’•๐’‚๐’ ๐’๐’–๐’Ž๐’ƒ๐’†๐’“ ๐’๐’‡ ๐’˜๐’‚๐’š๐’”๏ทฏ Total number of men= 2 Number of men to be selected = 2 Number of ways to select 2 men = 2C2 = ๏ท2!๏ทฎ2!๏ท2 โˆ’2๏ทฏ!๏ทฏ = ๏ท2!๏ทฎ2!0!๏ทฏ = ๏ท2!๏ทฎ2! ร— 1๏ทฏ = 1 ways P(two men are selected) = ๏ท๐‘๐‘ข๐‘š๐‘๐‘’๐‘Ÿ ๐‘œ๐‘“ ๐‘ค๐‘Ž๐‘ฆ๐‘  ๐‘๐‘œ๐‘กโ„Ž ๐‘š๐‘’๐‘› ๐‘–๐‘  ๐‘ ๐‘’๐‘™๐‘’๐‘๐‘ก๐‘’๐‘‘๏ทฎ๐‘‡๐‘œ๐‘ก๐‘Ž๐‘™ ๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ ๐‘œ๐‘“ ๐‘ค๐‘Ž๐‘ฆ๐‘ ๏ทฏ = ๏ท๐Ÿ๏ทฎ๐Ÿ”๏ทฏ

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