Write down the value of k.

Show that μ = 106.

Find P(M < 95) .

Find the probability that a bag of apples selected at random contains at most one small apple.

Find the expected number of bags in this crate that contain at most one small apple.

Find the probability that at least 48 bags in this crate contain at most one small apple.

evidence of using \(\sum {{p_i}}  = 1\)     (M1)

eg   k + 0.98 + 0.01 = 1

k = 0.01     A1 N2

[2 marks]

recognizing that 93 and 119 are symmetrical about μ       (M1)

eg   μ is midpoint of 93 and 119

correct working to find μ       A1

\(\frac{{119 + 93}}{2}\)

μ = 106     AG N0

[2 marks]

finding standardized value for 93 or 119      (A1)
eg   z = −2.32634, z = 2.32634

correct substitution using their z value      (A1)
eg  \(\frac{{93 - 106}}{\sigma } =  - 2.32634,\,\,\frac{{119 - 106}}{{2.32634}} = \sigma \)

σ = 5.58815     (A1)

0.024508

P(X < 95) = 0.0245      A2 N3

[5 marks]

evidence of recognizing binomial    (M1) 

eg 10, ananaCpqn−=××and 0.024B(5,,)pnp= 

valid approach    (M1) 

eg P(1),P(0)P(1)XXX≤=+= 

0.976285 

0.976     A1 N2 

[3 marks]

recognizing new binomial probability      (M1)
eg     B(50, 0.976)

correct substitution      (A1)
eg     E(X) = 50 (0.976285)

48.81425

48.8    A1 N2

[3 marks]

valid approach      (M1)

eg   P(X ≥ 48), 1 − P(X ≤ 47)

0.884688

0.885       A1 N2

[2 marks]