Breaking/cutting problems

5. A piece of wood \(A B\) is 3 metres long. The wood is cut at random at a point \(C\) and the random variable \(W\) represents the length of the piece of wood \(A C\).

1. Find the probability that the length of the piece of wood \(A C\) is more than 1.8 metres. The two pieces of wood \(A C\) and \(C B\) form the two shortest sides of a right-angled triangle. The random variable \(X\) represents the length of the longest side of the right-angled triangle.
2. Show that \(X ^ { 2 } = 2 W ^ { 2 } - 6 W + 9\) [0pt] [You may assume for random variables \(S , T\) and \(U\) and for constants \(a\) and \(b\) that if \(S = a T + b U\) then \(\mathrm { E } ( S ) = a \mathrm { E } ( T ) + b \mathrm { E } ( U ) ]\)
3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
4. Find \(\mathrm { P } \left( X ^ { 2 } > 5 \right)\)

1. The random variable \(W\) has a continuous uniform distribution over the interval \([ - 6 , a ]\) where \(a\) is a constant.

Given that \(\operatorname { Var } ( W ) = 27\)

1. show that \(a = 12\) Given that \(\mathrm { P } ( W > b ) = \frac { 3 } { 5 }\)
2. 1. find the value of \(b\)
   2. find \(\mathrm { P } \left( - 12 < W < \frac { b } { 2 } \right)\) A piece of wood \(A B\) has length 160 cm . The wood is cut at random into 2 pieces. Each of the pieces is then cut in half. The four pieces are used to form the sides of a rectangle.
3. Calculate the probability that the area of the rectangle is greater than \(975 \mathrm {~cm} ^ { 2 }\)

2. (i) The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) Given that \(\mathrm { P } ( 8 < X < 14 ) = \frac { 1 } { 5 }\) and \(\mathrm { E } ( X ) = 11\)

1. write down \(\mathrm { P } ( X > 14 )\)
2. find \(\mathrm { P } ( 6 X > a + b )\) (ii) Susie makes a strip of pasta 45 cm long. She then cuts the strip of pasta, at a randomly chosen point, into two pieces. The random variable \(S\) is the length of the shortest piece of pasta.
   1. Write down the distribution of \(S\)
   2. Calculate the probability that the shortest piece of pasta is less than 12 cm long. Susie makes 20 strips of pasta, all 45 cm long, and separately cuts each strip of pasta, at a randomly chosen point, into two pieces.
   3. Calculate the probability that exactly 6 of the pieces of pasta are less than 12 cm long.

3. A rod of length \(2 l\) was broken into 2 parts. The point at which the rod broke is equally likely to be anywhere along the rod. The length of the shorter piece of rod is represented by the random variable \(X\).

1. Write down the name of the probability density function of \(X\), and specify it fully.
2. Find \(\mathrm { P } \left( X < \frac { 1 } { 3 } l \right)\).
3. Write down the value of \(\mathrm { E } ( X )\). Two identical rods of length \(2 l\) are broken.
4. Find the probability that both of the shorter pieces are of length less than \(\frac { 1 } { 3 } l\).

7. A piece of string \(A B\) has length 9 cm . The string is cut at random at a point \(P\) and the random variable \(X\) represents the length of the piece of string \(A P\).

1. Write down the distribution of \(X\).
2. Find the probability that the length of the piece of string \(A P\) is more than 6 cm . The two pieces of string \(A P\) and \(P B\) are used to form two sides of a rectangle. The random variable \(R\) represents the area of the rectangle.
3. Show that \(R = a X ^ { 2 } + b X\) and state the values of the constants \(a\) and \(b\).
4. Find \(\mathrm { E } ( R )\).
5. Find the probability that \(R\) is more than twice the area of a square whose side has the length of the piece of string \(A P\).

3. A string of length 60 cm is cut a random point.

1. Name a distribution, including parameters, that can be used to model the length of the longer piece of string and find its mean and variance.
2. The longer string is shaped to form the perimeter of a circle. Find the probability that the area of the circle is greater than \(100 \mathrm {~cm} ^ { 2 }\).

A piece of string \(AB\) has length 12 cm. A child cuts the string at a randomly chosen point \(P\), into two pieces. The random variable \(X\) represents the length, in cm, of the piece \(AP\).

1. Suggest a suitable model for the distribution of \(X\) and specify it fully [2]
2. Find the cumulative distribution function of \(X\). [4]
3. Write down P(\(X < 4\)). [1]

A child cuts a 30 cm piece of string into two parts, cutting at a random point.

1. Name the distribution of \(L\), the length of the longer part of string, and sketch the probability density function for \(L\). [4 marks]
2. Find the probability that one part of the string is more than twice as long as the other. [3 marks]