A tree is selected at random from the forest.

Find the probability that this tree is a giant.

A tree is selected at random from the forest.

Given that this tree is a giant, find the probability that it is taller than $70$ metres.

Two trees are selected at random. Find the probability that they are both giants.

$100$ trees are selected at random.

Find the expected number of these trees that are giants.

$100$ trees are selected at random.

Find the probability that at least $25$ of these trees are giants.

valid approach     (M1)

eg     ${\text{P}}(G) = {\text{P}}(H > 60,{\text{ }}z = 0.875,{\text{ P}}(H > 60) = 1 - 0.809,{\text{ N}}\left( {53, {8^2}} \right)$

$0.190786$

${\text{P}}(G) = 0.191$     A1     N2

[3 marks]

finding ${\text{P}}(H > 70) = 0.01679$  (seen anywhere)     (A1)

recognizing conditional probability     (R1)

eg     ${\text{P}}(A\left| {B),{\text{ P}}(H > 70\left| {H > 60)} \right.} \right.$

correct working     (A1)

eg     $\frac{{0.01679}}{{0.191}}$

$0.0880209$

${\text{P}}(X > 70\left| {G) = 0.0880} \right.$     A1     N3

[6 marks]

attempt to square their ${\text{P}}(G)$     (M1)

eg     ${0.191^2}$

$0.0363996$

${\text{P}}({\text{both }}G) = 0.0364$     A1     N2

[2 marks]

correct substitution into formula for ${\text{E}}(X)$     (A1)

eg     $100(0.191)$

${\text{E}}(G) = 19.1{\text{ }}[19.0,{\text{ }}19.1]$     A1     N2

[3 marks]

recognizing binomial probability (may be seen in part (c)(i))     (R1)

eg     $X \sim {\text{B}}(n,{\text{ }}p)$

valid approach (seen anywhere)     (M1)

eg     ${\text{P}}(X \geqslant 25) = 1 - {\text{P}}(X \leqslant 24),{\text{ }}1 - {\text{P}}(X < a)$

correct working     (A1)

eg     ${\text{P}}(X \leqslant 24) = 0.913 \ldots ,{\text{ }}1 - 0.913 \ldots$

$0.0869002$

${\text{P}}(X \geqslant 25) = 0.0869$     A1     N2

[3 marks]