The random variable \(X\) has the distribution \(\mathrm{B}(6, 0.25)\). Sketch the distribution of \(X\).[2] One of the jobs of a quality control operative in a food-processing factory is to check the quality of a random sample of \(50\) meat pies from the production line each day. She records the number of pies found to be 'unsatisfactory' each day for \(80\) days. Her results are shown in the table below. Number unsatisfactory\(0\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\) or more Frequency\(57\)\(9\)\(6\)\(4\)\(3\)\(1\)\(0\) Use the information in the table to estimate the probability that a randomly chosen pie is unsatisfactory. [2] One of the products made in the food-processing factory is the 'Frozen Cheesee Burger'. A fixed number of the burgers are tested each day and the number found to have insufficient cheese in them is denoted by \(Y\). State, in context, two assumptions needed for \(Y\) to be well modelled by a binomial distribution. [2] Assume now that \(Y\) has the distribution \(\mathrm{B}(120, 0.03)\). Find the probability that, on a randomly chosen day, fewer than \(3\) burgers are found to have insufficient cheese in them. [1] For a randomly chosen period of \(28\) days, find the expectation of the number of days on which fewer than \(3\) burgers have insufficient cheese in them. [2] Find the probability that, in a randomly chosen period of \(28\) days, more than \(100\) burgers are found to have insufficient cheese in them. [2]

N2024 P2 Q9

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The random variable \(X\) has the distribution \(\mathrm{B}(6, 0.25)\). Sketch the distribution of \(X\).[2]

One of the jobs of a quality control operative in a food-processing factory is to check the quality of a random sample of \(50\) meat pies from the production line each day. She records the number of pies found to be 'unsatisfactory' each day for \(80\) days. Her results are shown in the table below.
Number unsatisfactory \(0\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) or more
Frequency \(57\) \(9\) \(6\) \(4\) \(3\) \(1\) \(0\)
Use the information in the table to estimate the probability that a randomly chosen pie is unsatisfactory.
[2]
One of the products made in the food-processing factory is the 'Frozen Cheesee Burger'. A fixed number of the burgers are tested each day and the number found to have insufficient cheese in them is denoted by \(Y\).
1. State, in context, two assumptions needed for \(Y\) to be well modelled by a binomial distribution.
  [2]
Assume now that \(Y\) has the distribution \(\mathrm{B}(120, 0.03)\).
1. Find the probability that, on a randomly chosen day, fewer than \(3\) burgers are found to have insufficient cheese in them.
  [1]
2. For a randomly chosen period of \(28\) days, find the expectation of the number of days on which fewer than \(3\) burgers have insufficient cheese in them.
  [2]
3. Find the probability that, in a randomly chosen period of \(28\) days, more than \(100\) burgers are found to have insufficient cheese in them.
  [2]

Number unsatisfactory	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\) or more
Frequency	\(57\)	\(9\)	\(6\)	\(4\)	\(3\)	\(1\)	\(0\)

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