Question

A company produces plastic elephants in two colours for the novelty trade market. Production in the factory is on one of three machines; 10% is on machine A, 30% on machine B, and the remainder on machine C. Machine A’s production consists of 40% blue elephants and 60% pink elephants. Machine B’s production consists of 30% blue elephants and 70% pink elephants. Machine C’s production has 80% pink elephants with the remainder being blue.
2.1.1 What proportion do blue elephants form of total production? (5)
2.1.2 If a particular elephant is pink, what is the probability it was made by machine B? (5)
2.2 You have been given the probability distributions of possible profits from two projects, A and B:

Answer 1

Answer on Question #40849 – Math - Other

Question

A company produces plastic elephants in two colours for the novelty trade market. Production in the factory is on one of three machines; 10% is on machine A, 30% on machine B, and the remainder on machine C. Machine A's production consists of 40% blue elephants and 60% pink elephants. Machine B's production consists of 30% blue elephants and 70% pink elephants. Machine C's production has 80% pink elephants with the remainder being blue.

2.1.1 What proportion do blue elephants form of total production?

2.1.2 If a particular elephant is pink, what is the probability it was made by machine B?

Solution

Brief

2.1.1.

$$T_B / T = 0.4 \cdot 0.1 + 0.3 \cdot 0.3 + 0.2 \cdot 0.6 = 0.25 \text{ or } 1 / 4,$$

where $T_B$ – total production of blue elephants, $T$ – total production.

2.1.2.

$$P = B_P / T_P = 0.7 \cdot 0.3 / (0.6 \cdot 0.1 + 0.7 \cdot 0.3 + 0.8 \cdot 0.6) = 0.28,$$

where $P$ – the probability, $T_P$ – total production of pink elephants, $B_P$ – production of pink elephants on machine B.

Detailed

Let's assign $T$ – total production of the elephants, $A$ – production on machine A, $B$ – production on machine B, $C$ – production on machine C.

The production of the elephants on each particular machine:

$$A = T \cdot 10\% / 100\% = 0.1 \cdot T$$ $$B = T \cdot 30\% / 100\% = 0.3 \cdot T$$ $$C = T \cdot (100\% - 10\% - 30\%) / 100\% = T \cdot 60\% / 100\% = 0.6 \cdot T$$

The production of blue elephants on each particular machine:

$$A_B = A \cdot 40\% / 100\% = 0.4 \cdot A = 0.4 \cdot 0.1 \cdot T = 0.04 \cdot T$$ $$B_B = B \cdot 30\% / 100\% = 0.4 \cdot B = 0.3 \cdot 0.3 \cdot T = 0.09 \cdot T$$ $$C_B = C \cdot (100\% - 80\%) / 100\% = C \cdot 20\% / 100\% = 0.2 \cdot C = 0.2 \cdot 0.6 \cdot T = 0.12 \cdot T$$

The production of pink elephants on each particular machine:

$$A_P = A \cdot 60\% / 100\% = 0.6 \cdot A = 0.6 \cdot 0.1 \cdot T = 0.06 \cdot T$$ $$B_P = B \cdot 70\% / 100\% = 0.7 \cdot B = 0.7 \cdot 0.3 \cdot T = 0.21 \cdot T$$ $$C_P = C \cdot 80\% / 100\% = 0.8 \cdot C = 0.8 \cdot 0.6 \cdot T = 0.48 \cdot T$$

Total production of blue elephants:

$$T_B = A_B + B_B + C_B = 0.04 \cdot T + 0.09 \cdot T + 0.12 \cdot T = (0.04 + 0.09 + 0.12) \cdot T = 0.25 \cdot T$$

So, the proportion of blue elephants of total production:

$$T_B / T = 0.25 \cdot T / T = 0.25 \text{ or } 1 / 4$$

Total production of pink elephants:

$$T_P = A_P + B_P + C_P = 0.06 \cdot T + 0.21 \cdot T + 0.48 \cdot T = (0.06 + 0.21 + 0.48) \cdot T = 0.75 \cdot T$$

The probability the particular pink elephant was made by machine B:

$$P = B_P / T_P = 0.21 \cdot T / 0.75 \cdot T = 0.28$$

Answers

2.1.1} 0.25

2.1.2} 0.28