(a) In a bakery, 30% of loaves of bread produced are of bad quality. If twelve loaves are selected at random from the bakery, calculate, correct to four dec...

(a) In a bakery, 30% of loaves of bread produced are of bad quality. If twelve loaves are selected at random from the bakery, calculate, correct to four decimal places. the probabshty of getting 

(i) exactly 6 bad ones:

(ii) at least 4 bad ones;

(ii) no bad one.

(b) A group consists of 8 boys and 5 girls. A committee of 7 members is chosen from the group. Find the probability that the committee is made up of 4 boys and 3 girls. 
 

Answer Details

(a) In this problem, we are dealing with a binomial distribution since we are selecting loaves of bread with replacement and there are only two possible outcomes: bad or good.

1. (i) The probability of getting exactly 6 bad loaves can be calculated using the binomial probability formula:
   P(X = 6) = (12 choose 6) * (0.3)^6 * (0.7)^6
   where "12 choose 6" is the number of ways to choose 6 loaves out of 12. Using a calculator, we get:
   P(X = 6) ≈ 0.2120
2. (ii) The probability of getting at least 4 bad loaves can be calculated by adding the probabilities of getting 4, 5, 6, ..., or 12 bad loaves:
   P(X ≥ 4) = P(X = 4) + P(X = 5) + ... + P(X = 12)
   Using the binomial probability formula for each term and a calculator, we get:
   P(X ≥ 4) ≈ 0.7094
3. (iii) The probability of getting no bad loaves can be calculated using the binomial probability formula for X = 0:
   P(X = 0) = (12 choose 0) * (0.3)^0 * (0.7)^12
   Using a calculator, we get:
   P(X = 0) ≈ 0.0138

(b) In this problem, we are dealing with a hypergeometric distribution since we are selecting a committee without replacement from a group with a fixed number of boys and girls.

The number of ways to choose 4 boys out of 8 is (8 choose 4), and the number of ways to choose 3 girls out of 5 is (5 choose 3). The total number of ways to choose 7 members out of 13 is (13 choose 7).

Therefore, the probability of choosing a committee with 4 boys and 3 girls is:

(8 choose 4) * (5 choose 3) / (13 choose 7) ≈ 0.2593

So the probability of choosing a committee with 4 boys and 3 girls is approximately 0.2593.