Broken Egg Probability

Explanation

Let's define the events as follows:

• $S$ = Sample space (total eggs)
• $D$ = Event of picking a duck egg
• $C$ = Event of picking a chicken egg
• $B$ = Event of picking a broken egg

First, we calculate the total number of eggs in the basket:

Next, we calculate the number of broken eggs:

• Broken duck eggs: $\frac{1}{2} \times 10 = 5$
• Broken chicken eggs: $\frac{1}{4} \times 20 = 5$

The total number of broken eggs ($n(B)$) is:

We are asked to find the probability of picking a duck egg or a broken egg ($D \cup B$). The formula for the probability of the union of two events is:

Where $n(D \cup B)$ can be calculated using the inclusion-exclusion principle:

• $n(D) = 10$ (number of duck eggs)
• $n(B) = 10$ (total number of broken eggs)
• $n(D \cap B)$ is the number of eggs that are both duck eggs and broken. From the calculation above, there are $5$ broken duck eggs.

Thus:

So the probability is:

Therefore, the probability of picking a duck egg or a broken egg is $0.5$.