The first plant supplies 30% of the CRTs, the second plant 40%, the third 30%. The first plant produces

The first plant supplies 30% of the CRTs, the second plant 40%, the third 30%. The first plant produces 80% of standard picture tubes, the second plant – 70%, the third – 85%. What is the probability that the picture tube taken at random is standard?

1. Consider the hypotheses for each CRT manufacturer:

A1 – CRT was produced at the first plant;
A2 – at the second plant;
A3 – at the third plant;
P (A1) = 0.3;
P (A2) = 0.4;
P (A3) = 0.3.
2. Conditional probabilities of the event X, consisting in the fact that the kinescope is standard:

P (X | A1) = 0.8;
P (X | A2) = 0.7;
P (X | A3) = 0.85.
3. According to the formula for the total probability we get:

P (X) = P (A1) * P (X | A1) + P (A2) * P (X | A2) + P (A3) * P (X | A3);
P (X) = 0.3 * 0.8 + 0.4 * 0.7 + 0.3 * 0.85 = 0.775.
Answer: 0.775.