Direct comparison with scalar multiple (different variables)

2 The random variable \(X\) has the distribution \(\mathrm { N } \left( 31.2,10.4 ^ { 2 } \right)\). Two independent random values of \(X\), denoted by \(X _ { 1 }\) and \(X _ { 2 }\), are chosen. Find \(\mathrm { P } \left( X _ { 1 } > 3 X _ { 2 } \right)\).

4 A random variable \(X\) has the distribution \(\mathrm { N } ( 10,12 )\). Two independent values of \(X\), denoted by \(X _ { 1 }\) and \(X _ { 2 }\), are chosen at random.

1. Write down the value of \(\mathrm { P } \left( X _ { 1 } > X _ { 2 } \right)\).
2. Find \(\mathrm { P } \left( X _ { 1 } > 2 X _ { 2 } - 3 \right)\).

6 The volumes, in millilitres, of large and small cups of tea are modelled by the distributions \(\mathrm { N } ( 200,30 )\) and \(\mathrm { N } ( 110,20 )\) respectively.

1. Find the probability that the total volume of a randomly chosen large cup of tea and a randomly chosen small cup of tea is less than 300 ml .
2. Find the probability that the volume of a randomly chosen large cup of tea is more than twice the volume of a randomly chosen small cup of tea.

4 The heights of a certain variety of plant are normally distributed with mean 110 cm and variance \(1050 \mathrm {~cm} ^ { 2 }\). Two plants of this variety are chosen at random. Find the probability that the height of one of these plants is at least 1.5 times the height of the other.

4 The masses, in grams, of large bags of sugar and small bags of sugar are denoted by \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} \left( 5.1,0.2 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 2.5,0.1 ^ { 2 } \right)\). Find the probability that the mass of a randomly chosen large bag is less than twice the mass of a randomly chosen small bag.

3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. \includegraphics[max width=\textwidth, alt={}, center]{65b50bfb-5fd8-4cf3-ae3b-cffc12e23cd8-06_76_1659_484_244}

3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. \includegraphics[max width=\textwidth, alt={}, center]{6346fd4b-7bc9-4205-94db-67368b9415fe-06_76_1659_484_244}

3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. \includegraphics[max width=\textwidth, alt={}, center]{fb305858-2d96-4a5d-b1a9-a965c248fb8d-06_76_1659_484_244}

3 The masses, in kilograms, of female and male animals of a certain species have the distributions \(\mathrm { N } \left( 102,27 ^ { 2 } \right)\) and \(\mathrm { N } \left( 170,55 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen female has a mass that is less than half the mass of a randomly chosen male. \includegraphics[max width=\textwidth, alt={}, center]{937c15d2-fb12-4af8-96d3-c54c81d771ba-06_76_1659_484_244}

7 Kieran and Andreas are long-jumpers. They model the lengths, in metres, that they jump by the independent random variables \(K \sim \mathrm {~N} ( 5.64,0.0576 )\) and \(A \sim \mathrm {~N} ( 4.97,0.0441 )\) respectively. They each make a jump and measure the length. Find the probability that

1. the sum of the lengths of their jumps is less than 11 m ,
2. Kieran jumps more than 1.2 times as far as Andreas.

1 The masses, in grams, of potatoes of types \(A\) and \(B\) have the distributions \(\mathrm { N } \left( 175,60 ^ { 2 } \right)\) and \(\mathrm { N } \left( 105,28 ^ { 2 } \right)\) respectively. Find the probability that a randomly chosen potato of type \(A\) has a mass that is at least twice the mass of a randomly chosen potato of type \(B\).

7. A set of scaffolding poles come in two sizes, long and short. The length \(L\) of a long pole has the normal distribution \(\mathrm { N } \left( 19.7,0.5 ^ { 2 } \right)\). The length \(S\) of a short pole has the normal distribution \(\mathrm { N } \left( 4.9,0.2 ^ { 2 } \right)\). The random variables \(L\) and \(S\) are independent. A long pole and a short pole are selected at random.

1. Find the probability that the length of the long pole is more than 4 times the length of the short pole. Four short poles are selected at random and placed end to end in a row. The random variable \(T\) represents the length of the row.
2. Find the distribution of \(T\).
3. Find \(\mathrm { P } ( | L - T | < 0.1 )\).

The masses, in kilograms, of chemicals \(A\) and \(B\) produced per day by a factory are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim\) N(10.3, 5.76) and \(Y \sim\) N(11.4, 9.61). The income generated by the chemicals is \\(2.50 per kilogram for \)A\( and \\)3.25 per kilogram for \(B\).

1. Find the mean and variance of the daily income generated by chemical \(A\). [2]
2. Find the probability that, on a randomly chosen day, the income generated by chemical \(A\) is greater than the income generated by chemical \(B\). [6]

Alun does the crossword in the Daily Bugle every day. The time that he takes to complete the crossword, \(X\) minutes, is modelled by the normal distribution \(\mathrm{N}(32, 4^2)\). You may assume that the times taken to complete the crossword on successive days are independent.

1. 1. Find the upper quartile of \(X\) and explain its meaning in context.
   2. Find the probability that the total time taken by Alun to complete the crosswords on five randomly chosen days is greater than 170 minutes. [7]
2. Belle also does the crossword every day and the time that she takes to complete the crossword, \(Y\) minutes, is modelled by the normal distribution \(\mathrm{N}(18, 2^2)\). Find the probability that, on a randomly chosen day, the time taken by Alun to complete the crossword is more than twice the time taken by Belle to complete the crossword. [6]

The length \(M\) of male snakes of a certain species may be regarded as a normal random variable with mean \(0.45\) metres and standard deviation \(0.06\) metres. The length \(F\) of female snakes of the same species may be regarded as a normal random variable with mean \(0.55\) metres and standard deviation \(0.08\) metres. Assuming that \(M\) and \(F\) are independent, find the probability that a randomly chosen male snake of this species is more than three-quarters of the length of a randomly chosen female snake of this species. [6]