(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.
Explanation
(a)P(Bad Quality) = (frac{30}{10}) = 0.3 and P(Good Quality) = 1 – 0.3 = 0.7
(a)(i) P(exactly 6) = (^{12}C_6) x (0.3)(^6) x (0.7)(^6) = 0.0792
(a)(ii) P(at least 4) = 1 – [P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3)]
Where P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) = (^{12}C_0)(0.3)(^6)(0.7)(^{12}) + (^{12}C_1)(0.3)(^1)(0.7)(^{11}) + (^{12}C_2)(0.3)(^2)(0.7)(^{10}) + (^{12}C_3)(0.3)(^3)(0.7)(^{9})
= 0.01384 + 0.071184 + 0.16779 + 0.2397
Therefore; P(at least 4) = 1 – 0.4925 = 0.5075
(a)(iii) P(no bad one) = (^{12}C_0)(0.3)(^o)(0.7)(^{12}) = 0.0138
(b) To find P(4 boys and 3 girls) = (frac{^8C_4 times ^5C_3}{^{13}C_7})
= (frac{70 times 10}{1716})
= (frac{175}{429})
= 0.4079