Example 14

Transcript

Example 14 On her vacations Veena visits four cities (A, B, C and D) in a random order. What is the probability that she visits A before B? 4 cities can be visited in any of following order S = ﷐﷐ABCD, ABDC, ACBD, ACDB, ADBC, ADCB,﷮ BACD, BADC, BDAC, BDCA, BCAD, BCDA,﷮ CABD, CADB, CBDA, CBAD, CDAB, CDBA,﷮ DABC, DACB, DBCA, DBAC, DCAB, DCBA﷯﷯ n(S) = 24 Let E be the event that “she visits A before B“ Hence , E = ﷐﷐ABCD, ABDC, ADBC, ACDB, ADBC, ADCB,﷮CABD, CADB, CDAB, ﷮DABC, DACB, DCAB, ﷯﷯ Hence , E = ﷐﷐ABCD, ABDC, ADBC, ACDB, ADBC, ADCB,﷮CABD, CADB, CDAB, ﷮DABC, DACB, DCAB, ﷯﷯ n(E) = 12 P(E) = ﷐𝑛(𝐸)﷮𝑛(𝑆)﷯ = ﷐12﷮24﷯ = ﷐𝟏﷮𝟐﷯ Example 14 What is the probability that she visits (ii) A before B and B before C? S = ﷐﷐ABCD, ABDC, ACBD, ACDB, ADBC, ADCB,﷮ BACD, BADC, BDAC, BDCA, BCAD, BCDA,﷮ CABD, CADB, CBDA, CBAD, CDAB, CDBA,﷮ DABC, DACB, DBCA, DBAC, DCAB, DCBA﷯﷯ Let F be the event “she visits A before B and B before C “ F = ﷐﷐ABCD, ABDC, ADBC , DABC﷯﷯ So, n(F) = 4 P(F) = ﷐𝑛(𝐹)﷮𝑛(𝑆)﷯ = ﷐4﷮24﷯ = ﷐𝟏﷮𝟔﷯ Example 14 What is the probability that she visits (iii) A first and B last? S = ﷐﷐ABCD, ABDC, ACBD, ACDB, ADBC, ADCB,﷮ BACD, BADC, BDAC, BDCA, BCAD, BCDA,﷮ CABD, CADB, CBDA, CBAD, CDAB, CDBA,﷮ DABC, DACB, DBCA, DBAC, DCAB, DCBA﷯﷯ Let G be the event “she visits A first and B last” G = ﷐﷐ACDB, ADCB﷯﷯ So, n(G) = 2 P(G) = ﷐𝑛(𝐺)﷮𝑛(𝑆)﷯ = ﷐2﷮24﷯ = ﷐𝟏﷮𝟏𝟐﷯ Example 14 What is the probability that she visits (iv) A either first or second? S = ﷐﷐ABCD, ABDC, ACBD, ACDB, ADBC, ADCB,﷮ BACD, BADC, BDAC, BDCA, BCAD, BCDA,﷮ CABD, CADB, CBDA, CBAD, CDAB, CDBA,﷮ DABC, DACB, DBCA, DBAC, DCAB, DCBA﷯﷯ Let H be the event “she visits A either first or second” H = ﷐﷐ABCD, ABDC, ADBC, ACDB, ADBC, ADCB,﷮ BACD, BADC,CABD, CADB,DABC, DACB,﷯﷯ So, n(H) = 12 P(H) = ﷐𝑛(𝐻)﷮𝑛(𝑆)﷯ = ﷐12﷮24﷯ = ﷐𝟏﷮𝟐﷯ Example 14 What is the probability that she visits (v) A just before B? S = ﷐﷐ABCD, ABDC, ACBD, ACDB, ADBC, ADCB,﷮ BACD, BADC, BDAC, BDCA, BCAD, BCDA,﷮ CABD, CADB, CBDA, CBAD, CDAB, CDBA,﷮ DABC, DACB, DBCA, DBAC, DCAB, DCBA﷯﷯ Let I be the event “she visits A just before B” I = ﷐﷐ABCD, ABDC,CABD,CDAB,DABC,DCAB,﷯﷯ So, n(I) = 6 P(I) = ﷐𝑛(𝐼)﷮𝑛(𝑆)﷯ = ﷐6﷮24﷯ = ﷐𝟏﷮𝟒﷯