Answer to Question #340126 in Statistics and Probability for Maneo

Question

There are 9 eggs out of which 4 are bad. 3 eggs are chosen at random.

1) In how many ways can we choose the 3 eggs from among the 9 eggs?

2) In how many ways can we choose 3 good eggs?

3) What is the probability that all selected eggs are good?

4) In how many ways can we choose 3 bad eggs?

5) What is the probability that all 3 selected eggs are bad ones?

6) What is the number of pairs of good eggs?

7) In how many ways can we choose 3 eggs out of which 1 is bad?

8) What is the probability that exactly one egg is bad?

9) What is the probability that at least one egg is bad?

Answer 1

1)

$$\dbinom{9}{3}=84$$

2)

$$\dbinom{4}{0}\dbinom{9-4}{3}=1(10)=10$$

3)

$$P(3\ good)=\dfrac{10}{84}=\dfrac{5}{42}$$

4)

$$\dbinom{4}{3}\dbinom{9-4}{0}=4(1)=4$$

5)

$$P(3\ bad)=\dfrac{4}{84}=\dfrac{1}{21}$$

6)

$$\dbinom{9-4}{2}=10$$

7)

$$\dbinom{4}{1}\dbinom{9-4}{2}=4(10)=40$$

8)

$$P(2\ good \ \&\ 1\ bad)=\dfrac{40}{84}=\dfrac{10}{21}$$

9)

$$P(at\ least\ 1\ bad)=1-\dfrac{5}{42}=\dfrac{37}{42}$$