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Chapter 7: Q29E (page 451)

Question:In a super lottery, a player wins a fortune if they choose the eight numbers selected by a computer from the positive integers not exceeding 100. What is the probability that a player wins this super lottery?

Short Answer

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Answer

The probability that a person wins the grand prize by picking 7 numbers that are among the 11 numbers selected at random by a computer, provided a player selects 7 numbers out of the first 80 positive integers is P(E)=5.37×10-12.

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01

 Given  

The positive integers not exceeding 100.

02

The Concept of Probability

IfSrepresents the sample space andE represents the event. Then the probability of occurrence of favourable event is given by the formula as below:

P(E)=n(E)n(S)

03

Determine the probability

As per the problem we have been asked to find the probability that a person wins the grand prize by picking 8 numbers selected by a computer from the positive integers not exceeding 100.

If Srepresents the sample space and Erepresents the event. Then the probability of occurrence of favourable event is given by the formula as below:

P(E)=n(E)n(S).

So, the number of possible ways of select 8 numbers out of the first 100 positive integers isn(s)=100C8.

Total number of ways that the 8 numbers selected by a player to be in 8 numbers selected by computer at random isn(E)=8C8.

There will be only one set of positive integers for which lottery can be drawn.

The probability that a person wins the grand prize by picking 8 numbers selected by a computer from the positive integers not exceeding 100 is as follows:

P(E)=C88C8100P(E)=1186087894300P(E)=5.37×10-12

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Most popular questions from this chapter

Question: Prove Theorem \(2\), the extended form of Bayes’ theorem. That is, suppose that \(E\) is an event from a sample space \(S\) and that \({F_1},{F_2},...,{F_n}\) are mutually exclusive events such that \(\bigcup\nolimits_{i = 1}^n {{F_i} = S} \). Assume that \(p\left( E \right) \ne 0\) and \(p\left( {{F_i}} \right) \ne 0\) for \(i = 1,2,...,n\). Show that

\(p\left( {{F_j}\left| E \right.} \right) = \frac{{p\left( {E\left| {{F_j}} \right.} \right)p\left( {{F_j}} \right)}}{{\sum\nolimits_{i = 1}^n {p\left( {E\left| {{F_i}} \right.} \right)p\left( {{F_i}} \right)} }}\)

(Hint: use the fact that \(E = \bigcup\nolimits_{i = 1}^n {\left( {E \cap {F_i}} \right)} \).)

Question: Assume that the probability a child is a boy is 0.51 and that the sexes of the children born into a family are independent. What is the probability that a family of five children has

  1. Exactly three boys?
  2. At least one boy?
  3. At least one girl?
  4. All children of the same sex?

Question: Suppose that a Bayesian spam filter is trained on a set of 1000 spam messages and 400 messages that are not spam. The word “opportunity” appears in 175 spam messages and 2 messages that are not spam. Would an incoming message be rejected as spam if it contains the word “opportunity” and the threshold for rejecting a message is 0.9?

Suppose that Ann selects a ball by first picking one of two boxes at random and then selecting a ball from this box. The first box contains three orange balls and four black balls, and the second box contains five orange balls and six black balls. What is the probability that Ann picked a ball from the second box if she has selected an orange ball?

Question: Suppose that we have prior information concerning whether a random incoming message is spam. In particular, suppose that over a time period, we find that s spam messages arrive and h messages arrive that are not spam.

a) Use this information to estimate p(S), the probability that an incoming message is spam, and\(p(\bar S)\), the probability an incoming message is not spam.

b) Use Bayes’ theorem and part (a) to estimate the probability that an incoming message containing the word w is spam, where p(w)is the probability that w occurs in a spam message and q(w) is the probability that w occurs in a message that is not spam.
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