a) Use normal distribution:

\( C \sim N(62, 2.9^2) \)

\( z = \frac{61 – 62}{2.9} = \frac{-1}{2.9} \approx -0.3448 \)

\( P(Z < -0.3448) \approx 1 – P(Z < 0.3448) \)

\( P(Z < 0.3448) \approx 0.6352 \)

\( P(C < 61) \approx 1 – 0.6352 = 0.3648 \)

Rounded to three decimal places:

\( \approx 0.365 \) [3]

b) Use binomial distribution:

\( X \sim B(12, 0.365) \)

\( P(X = 5) = \binom{12}{5} \times 0.365^5 \times 0.635^7 \)

\( \binom{12}{5} = 792 \)

\( 0.365^5 \approx 0.00064757 \)

\( 0.635^7 \approx 0.041595 \)

\( P(X = 5) \approx 792 \times 0.00064757 \times 0.041595 \approx 0.21351 \)

Rounded to three decimal places:

\( \approx 0.214 \) [3]

ci) Use total probability:

\( B \sim N(68, 3.4^2) \)

\( z = \frac{61 – 68}{3.4} = \frac{-7}{3.4} \approx -2.0588 \)

\( P(Z < -2.0588) \approx 0.01977 \)

\( P(\text{Chocolate}) = 0.6 \), \( P(\text{Banana}) = 0.4 \)

\( P(C < 61) \approx 0.365 \)

\( P(B < 61) \approx 0.01977 \)

\( P(\text{<61g}) = 0.6 \times 0.365 + 0.4 \times 0.01977 \)

\( \approx 0.219 + 0.007908 \approx 0.226908 \)

Rounded to three decimal places:

\( \approx 0.227 \) [3]

cii) Use conditional probability:

\( P(\text{Chocolate} | \text{<61g}) = \frac{P(\text{Chocolate} \cap \text{<61g})}{P(\text{<61g})} \)

\( P(\text{Chocolate} \cap \text{<61g}) = 0.6 \times 0.365 \approx 0.219 \)

\( P(\text{<61g}) \approx 0.226908 \)

\( P(\text{Chocolate} | \text{<61g}) \approx \frac{0.219}{0.226908} \approx 0.96525 \)

Rounded to three decimal places:

\( \approx 0.965 \) [3]

d) Use total probability with new standard deviation:

\( C \sim N(62, \sigma^2) \)

\( P(\text{<61g}) = 0.6 \times P(C < 61) + 0.4 \times 0.01977 = 0.157 \)

\( P(B < 61) \approx 0.01977 \)

\( 0.6 \times P(C < 61) + 0.007908 \approx 0.157 \)

\( 0.6 \times P(C < 61) \approx 0.149092 \)

\( P(C < 61) \approx \frac{0.149092}{0.6} \approx 0.248487 \)

\( z = \frac{61 – 62}{\sigma} = \frac{-1}{\sigma} \)

\( P\left(Z < \frac{-1}{\sigma}\right) = 0.248487 \)

\( P(Z < -z) = 0.751513 \)

\( -z \approx -0.6792 \)

\( z \approx 0.6792 \)

\( \frac{-1}{\sigma} = -0.6792 \)

\( \sigma \approx \frac{1}{0.6792} \approx 1.47214 \)

Rounded to two decimal places:

\( \sigma \approx 1.47 \) g [4]