Question: Joy and her best friend wanted to paint their apartment. In their storage closet, they found 8 cans of paint. Joy remembered that 5 of the cans contained yellow paint, but none of the cans had labels.
If Joy randomly opened 4 cans, what is the probability that all of them contained yellow paint?
Write your answer as a decimal rounded to four decimal places.
1. **State the problem:**
We have 8 cans of paint, 5 of which are yellow. Joy randomly opens 4 cans. We want to find the probability that all 4 opened cans are yellow.
2. **Formula and rules:**
The probability of selecting 4 yellow cans out of 5 yellow cans from a total of 8 cans is given by the ratio of combinations:
$$\text{Probability} = \frac{\binom{5}{4}}{\binom{8}{4}}$$
where $\binom{n}{k}$ is the number of ways to choose $k$ items from $n$ items.
3. **Calculate combinations:**
$$\binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5!}{4!1!} = 5$$
$$\binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} = 70$$
4. **Calculate probability:**
$$\text{Probability} = \frac{5}{70} = \frac{\cancel{5}}{\cancel{70}} = \frac{1}{14} \approx 0.0714$$
5. **Final answer:**
The probability that all 4 cans opened are yellow is approximately **0.0714**.
This means there is about a 7.14% chance that all 4 randomly opened cans contain yellow paint.