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?

Answer 1

Answer on Question #42681-Math- Statistics and Probability

A company produces plastic elephants in two colors 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.

$$\frac {T _ {B}}{T} = 0.4 \cdot 0.1 + 0.3 \cdot 0.3 + 0.2 \cdot 0.6 = 0.25,$$

where $T_{B}$ is total production of blue elephants, $T$ is total production.

2.1.2.

$$P = \frac {B _ {P}}{T _ {P}} = \frac {0.7 \cdot 0.3}{0.6 \cdot 0.1 + 0.7 \cdot 0.3 + 0.8 \cdot 0.6} = 0.28,$$

where $P$ is the probability, $T_P$ is total production of pink elephants, $B_P$ is 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:

$$A = T \frac {10\%}{100\%} = 0.1 \cdot T, \qquad B = T \frac {30\%}{100\%} = 0.3 \cdot T, \qquad C = T \frac {100\% - 30\% - 10\%}{100\%} = 0.6 \cdot T.$$

The production of blue elephants on each particular machine:

$$\begin{array}{l} A _ {B} = A \frac {40\%}{100\%} = 0.4 \cdot A = 0.04 \cdot T, \quad B _ {B} = B \frac {30\%}{100\%} = 0.3 \cdot B = 0.09 \cdot T, \\ C _ {B} = C \frac {100\% - 80\%}{100\%} = 0.2 \cdot C = 0.12 \cdot T. \end{array}$$

The production of pink elephants on each particular machine:

$$\begin{array}{l} A _ {P} = A \frac {60\%}{100\%} = 0.6 \cdot A = 0.06 \cdot T, \quad B _ {P} = B \frac {70\%}{100\%} = 0.7 \cdot B = 0.21 \cdot T, \\ C _ {P} = C \frac {80\%}{100\%} = 0.8 \cdot C = 0.48 \cdot T. \end{array}$$

Total production of blue elephants:

$$T _ {B} = A _ {B} + B _ {B} + C _ {B} = 0.04 \cdot T + 0.09 \cdot T + 0.12 \cdot T = 0.25 \cdot T.$$

So, the proportion of blue elephants of total production:

$$\frac {T _ {B}}{T} = \frac {0 . 2 5 \cdot T}{T} = 0. 2 5.$$

Total production of pink elephants:

$$T _ {P} = A _ {P} + B _ {P} + C _ {P} = 0. 0 6 \cdot T + 0. 2 1 \cdot T + 0. 4 8 \cdot T = 0. 7 5 \cdot T.$$

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

$$P = \frac {B _ {P}}{T _ {P}} = \frac {0 . 2 1 \cdot T}{0 . 7 5 \cdot T} = 0. 2 8.$$

Answer: 2.1.1) 0.25; 2.1.2) 0.28.

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