Elizabeth's Bakery makes brownies - AQA - A-Level Maths Mechanics - Question 17 - 2019 - Paper 3

To find $P(X < 35)$, we first calculate the z-score:

$z = \frac{35 - \mu}{\sigma} = \frac{35 - 37.3}{5.68} \approx -0.40$

Using the standard normal distribution table, we find:

$P(Z < -0.40) \approx 0.344\implies P(X < 35) \approx 0.344$

We know that:

• Let $Y$ be the number of brownies less than 35 grams in a batch of 13.
• $Y \sim \text{Binomial}(n=13, p=0.344)$, where $p$ is the probability of a brownie being less than 35 grams.

We are to find:

$P(Y \leq 3) = P(Y = 0) + P(Y = 1) + P(Y = 2) + P(Y = 3)$

Calculating individual probabilities using the binomial formula:

$P(Y = k) = \binom{n}{k} p^k (1-p)^{n-k}$

1. For $k=0$: $P(Y = 0) = \binom{13}{0} (0.344)^0 (0.656)^{13} \approx 0.022$
2. For $k=1$: $P(Y = 1) = \binom{13}{1} (0.344)^1 (0.656)^{12} \approx 0.098$
3. For $k=2$: $P(Y = 2) = \binom{13}{2} (0.344)^2 (0.656)^{11} \approx 0.199$
4. For $k=3$: $P(Y = 3) = \binom{13}{3} (0.344)^3 (0.656)^{10} \approx 0.227$

Finally, summing these probabilities gives:

$P(Y \leq 3) \approx 0.022 + 0.098 + 0.199 + 0.227 \approx 0.546$

Thus, the probability that no more than 3 brownies are less than 35 grams is approximately 0.546.