The probability of rejection for each product is 0.2. What is the probability that out of six selected products

The probability of rejection for each product is 0.2. What is the probability that out of six selected products, the number of non-defective products will be at least three?

Let q be the probability that the next product will be defective. By the condition of the problem, q = 0.2.

Let p be the probability that the next product will be of high quality (not defective).

p = 1 – 0.2 = 0.8.

Let’s say at least three products are of high quality. Then three, four, five or all six products may be of high quality.

Let’s find the probability that exactly three products will be of high quality. To do this, we will use the Bernoulli formula.

P6 (3) = C36 * p ^ 3 * q ^ (6 – 3);

C36 = 6! / (3! * (6 – 3)!) = 20;

P6 (3) = 20 * p ^ 3 * q ^ (6 – 3) = 20 * 0.8 ^ 3 * 0.2 ^ 3 = 0.08192.

Let’s find the probability that exactly four products will be of high quality. To do this, we will use the Bernoulli formula.

P6 (4) = C46 * p ^ 4 * q ^ (6 – 4);

C46 = 6! / (4! * (6 – 4)!) = 15;

P6 (4) = 15 * p ^ 4 * q ^ 2 = 15 * 0.8 ^ 4 * 0.2 ^ 2 = 0.24576.

Let’s find the probability that exactly five products will be of high quality. To do this, we will use the Bernoulli formula.

P6 (5) = C56 * p ^ 5 * q ^ (6 – 5);

C56 = 6! / (5! * (6 – 5)!) = 6;

P6 (5) = 6 * p ^ 5 * q ^ 1 = 6 * 0.8 ^ 5 * 0.2 = 0.393216.

Let’s find the probability that all six products will be of good quality.

P6 (6) = p ^ 6 = 0.8 ^ 6 = 0.262144.

Let’s find the probability that at least three products will be of high quality. To do this, add up the found probabilities.

0.08192 + 0.24576 + 0.393216 + 0.262144 = 0.98304 ≈ 98.3%.

Answer: approximately 98.3%.