Question

A fair die is rolled 1000 times. Let A be the event that the number of 6’s is in the interval[150,200], and B the event that the number of 5’s is exactly 200. (a) Approximate P(A).(b) Approximate P(A|B).

Answer #1

n= | 1000 | p=probability of a number= | 1/6 |

here mean of distribution=μ=np= | 166.67 | |

and standard deviation σ=sqrt(np(1-p))= | 11.79 | |

for normal distribution z score =(X-μ)/σx |

therefore from normal approximation of binomial distribution and continuity correction: |

a)

P(A)
=P(149.5<X<200.5)=P((149.5-166.67)/11.785)<Z<(200.5-166.67)/11.785)=P(-1.46<Z<2.87)=0.9979-0.0721=0.9258 |

b)

n= | 800 | p(A|B)=probability of 5 given 6 is not in them=1/5= | 0.20 |

here mean of distribution=μ=np= | 160.00 | |

and standard deviation σ=sqrt(np(1-p))= | 11.31 |

probability
=P(199.5<X<200.5)=P((199.5-160)/11.314)<Z<(200.5-160)/11.314)=P(3.49<Z<3.58)=0.9998-0.9997=0.0001 |

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Example 1 A fair six-sided die is rolled six times. If the face
numbered k is the outcome on roll k for k=1, 2, ..., 6, we say that
a match has occurred. The experiment is called a success if at
least one match occurs during the six trials. Otherwise, the
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Assign a value to P(A)
Simulate the experiment on...

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