To solve this problem, we will first compute the total number of possible outcomes, then list all outcomes in the sample space, and finally compute the probability that both chosen chips are defective.

Step 1: Total Possible Outcomes

We are selecting two chips out of six, without replacement. The total number of possible outcomes can be found using the combination formula:

$\text{Total outcomes} = \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15$

So, there are 15 possible pairs of chips that can be selected.

Step 2: Defining the Sample Space

We can represent the chips as:

• $D_1$ and $D_2$ for the two defective chips, and
• $G_1, G_2, G_3, G_4$ for the four good chips.

Now, we list all possible pairs of chips that can be selected. These are:

1. $(D_1, D_2)$
2. $(D_1, G_1)$
3. $(D_1, G_2)$
4. $(D_1, G_3)$
5. $(D_1, G_4)$
6. $(D_2, G_1)$
7. $(D_2, G_2)$
8. $(D_2, G_3)$
9. $(D_2, G_4)$
10. $(G_1, G_2)$
11. $(G_1, G_3)$
12. $(G_1, G_4)$
13. $(G_2, G_3)$
14. $(G_2, G_4)$
15. $(G_3, G_4)$

This is the complete sample space of all possible outcomes.

Step 3: Finding the Probability that Both Chosen Chips are Defective

There is only 1 outcome in which both chips are defective, which is $(D_1, D_2)$.

Thus, the probability that both selected chips are defective is:

$P(\text{both defective}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{15}$

Final Answer:

The probability that both chips chosen are defective is $\frac{1}{15}$.

Would you like further details or have any questions?

Related Questions:

1. What is the probability that neither of the chosen chips is defective?
2. What is the probability that at least one of the chosen chips is defective?
3. How does the probability change if three chips are chosen instead of two?
4. What is the probability of selecting one defective and one good chip?
5. How would the solution change if the chips were chosen with replacement?

Tip:

When solving problems involving combinations, it's useful to first define the total sample space clearly to make counting the favorable outcomes easier.