2024 YIJC P2 Q12

In this question you should state clearly all the distributions that you use, together with the values of the appropriate parameters.

A vegetable seller sells potatoes and tomatoes. The masses of potatoes, \(X\) grams, have the distribution \(N\left( 175,{{25}^{2}} \right)\).

1. Given that \(\mathrm{P}\!\left( X \lt 166 \right) = \mathrm{P}\!\left( X \gt m \right)\), find the value of \(m\).[1]
2. The probability that the mass of a randomly chosen potato differs from the population mean mass by more than \(a\) grams is \(0.3\). Find the value of \(a\).[2]
3. Find the expected number of potatoes with mass less than \(182\) grams in a randomly chosen batch of \(90\) potatoes.[2]

The masses of tomatoes, \(Y\) grams, have the distribution \(N\left( 125,{{15}^{2}} \right)\).

4. Find the probability that the mass of a randomly chosen tomato is more than \(130\) grams.[1]
5. A sample of \(50\) tomatoes are randomly chosen. Find the probability that exactly \(20\) tomatoes in this sample each has a mass more than \(130\)grams.[2]

The selling price of the potatoes is \(\$0.32\) per \(100\)g and the selling price of the tomatoes is \(\$0.22\) per \(100\)g.

6. Find the probability that the price of a randomly chosen potato is greater than the total price of \(2\) randomly chosen tomatoes.[4]