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If a die is rolled 40 times, there are 640 different sequences possible. The following question asks how many of these sequences satisfy certain conditions. HINT [Use the decision algorithm discussed in Example 3 of Section 6.3.] What fraction of these sequences have exactly five 1s? (Round your answer to four decimal places.)

Respuesta :

Answer:

0.1433

Step-by-step explanation:

All possibilities:

The total number of different sequences in 40 rolls of a die is [tex]6^{40}[/tex]

Exactly five 1s in 40 rolls:

                         The 1s may occur at any 5 rolls among 40. The number of ways of exactly five 1s occuring is therefore, equivalent to number of ways of selecting 5 fruits from 40 distinct fruits which is ⁴⁰C₅ ways. These 5 rolls have a fixed outcome 1. Other 35 rolls each have 5 possible outcomes : 2 or 3 or 4 or 5 or 6. So, the number of possible sequences of outcomes on other 35 rolls of the die is [tex]5^{35}[/tex].

⇒The total number of different sequences having exactly five 1s in 40 rolls of a die is (⁴⁰C₅)×([tex]5^{35}[/tex])

∴The fraction of the total number of sequences having exactly five 1s is [tex]\frac{(⁴⁰C₅)×([tex]5^{35}[/tex])}{[tex]6^{40}[/tex]}[/tex] ≈ 0.1433

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