Example 8 - Chapter 14 Class 11 Probability

Transcript

Example 8 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) = ﷐𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒘𝒂𝒚𝒔 𝒃𝒐𝒕𝒉 𝒘𝒐𝒎𝒆𝒏 𝒊𝒔 𝒔𝒆𝒍𝒆𝒄𝒕𝒆𝒅﷮𝑻𝒐𝒕𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒘𝒂𝒚𝒔﷯ Total Number of ways = 4C2 = ﷐4!﷮2!﷐4 −2﷯!﷯ = ﷐4!﷮2!﷐2﷯!﷯ = ﷐4 × 3 × 2!﷮2 × 1 × 2!﷯ = 6 Number of women = 2 Number of women to be selected = 2 Number of ways both women is selected = 2C2 = ﷐2!﷮2!﷐2 −2﷯!﷯ = ﷐2!﷮2!0!﷯ = ﷐2!﷮2! × 1﷯ = 1 ways Putting values in (1) P(no man is selected) = ﷐𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑏𝑜𝑡ℎ 𝑤𝑜𝑚𝑒𝑛 𝑖𝑠 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑﷮𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠﷯ = ﷐𝟏﷮𝟔﷯ Example 8 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) = ﷐𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒘𝒂𝒚𝒔 𝟏 𝒎𝒂𝒏 𝒊𝒔 𝒔𝒆𝒍𝒆𝒄𝒕𝒆𝒅 × 𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒘𝒂𝒚𝒔 𝟏 𝒘𝒐𝒎𝒂𝒏 𝒊𝒔 𝒔𝒆𝒍𝒆𝒄𝒕𝒆𝒅 ﷮𝑻𝒐𝒕𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒘𝒂𝒚𝒔﷯ Total Number of men = 2 Number of men to be selected = 1 Number of ways to select 1 man = 2C1 = ﷐2!﷮1!﷐2 −1﷯!﷯ = ﷐2!﷮1!﷯ = 2 ways Total Number of women = 2 Number of women to be selected = 1 Number of ways to select 1 woman = 2C1 = ﷐2!﷮1!﷐2 −1﷯!﷯ = ﷐2!﷮1!﷯ = 2 ways P(one man is selected) = ﷐𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 1 𝑚𝑎𝑛 𝑖𝑠 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑 × 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 1 𝑤𝑜𝑚𝑎𝑛 𝑖𝑠 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑 ﷮𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠﷯ = ﷐2 × 2﷮6﷯ = ﷐4﷮6﷯ = ﷐𝟐﷮𝟑﷯ Example 8 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) = ﷐𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒘𝒂𝒚𝒔 𝒃𝒐𝒕𝒉 𝒎𝒆𝒏 𝒊𝒔 𝒔𝒆𝒍𝒆𝒄𝒕𝒆𝒅﷮𝑻𝒐𝒕𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒘𝒂𝒚𝒔﷯ Total number of men= 2 Number of men to be selected = 2 Number of ways to select 2 men = 2C2 = ﷐2!﷮2!﷐2 −2﷯!﷯ = ﷐2!﷮2!0!﷯ = ﷐2!﷮2! × 1﷯ = 1 ways P(two men are selected) = ﷐𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑏𝑜𝑡ℎ 𝑚𝑒𝑛 𝑖𝑠 𝑠𝑒𝑙𝑒𝑐𝑡𝑒𝑑﷮𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠﷯ = ﷐𝟏﷮𝟔﷯