5 Most plants of a certain type have three leaves. However, it is known that, on average, 1 in 10000 of these plants have four leaves, and plants with four leaves are called 'lucky'. The number of lucky plants in a random sample of 25000 plants is denoted by \(X\).
- State, with a justification, an approximating distribution for \(X\), giving the values of any parameters.
Use your approximating distribution to answer parts (b) and (c). - Find \(\mathrm { P } ( X \leqslant 3 )\).
- Given that \(\mathrm { P } ( X = k ) = 2 \mathrm { P } ( X = k + 1 )\), find \(k\).
The number of lucky plants in a random sample of \(n\) plants, where \(n\) is large, is denoted by \(Y\). - Given that \(\mathrm { P } ( Y \geqslant 1 ) = 0.963\), correct to 3 significant figures, use a suitable approximating distribution to find the value of \(n\).