Matching Colors from Two Jars
Bob has two jars of marbles:
- Jar A: $4$ blue and $6$ red marbles ($10$ total)
- Jar B: $5$ blue and $3$ red marbles ($8$ total)
Bob draws one marble uniformly at random from each jar. What is the probability that the two marbles are the same color?
Hints
- The two draws are independent since they come from different jars -- matching means either both red or both blue.
- Compute each case separately: $P(\text{both red}) = P(\text{red from A}) \times P(\text{red from B})$, and similarly for blue.
- The probabilities are $P(\text{both red}) = (6/10)(3/8)$ and $P(\text{both blue}) = (4/10)(5/8)$. Add them up.
Worked Solution
How to Think About It: The two draws are independent (one from each jar), so we just need to add the probability of matching on red and the probability of matching on blue. This is a quick computation -- no tricks, just careful bookkeeping.
Quick Estimate: Jar A is $60\%$ red, Jar B is $37.5\%$ red. The probability of both red is roughly $0.6 \times 0.375 = 0.225$. Jar A is $40\%$ blue, Jar B is $62.5\%$ blue, so both blue is roughly $0.4 \times 0.625 = 0.25$. Total match probability is about $0.475$, so just under $50\%$.
Formal Solution:
The draws are independent, so:
$$P(\text{match}) = P(\text{both red}) + P(\text{both blue})$$
Both red:
$$P(\text{red from A}) \cdot P(\text{red from B}) = \frac{6}{10} \cdot \frac{3}{8} = \frac{18}{80} = \frac{9}{40}$$
Both blue:
$$P(\text{blue from A}) \cdot P(\text{blue from B}) = \frac{4}{10} \cdot \frac{5}{8} = \frac{20}{80} = \frac{10}{40}$$
Total:
$$P(\text{match}) = \frac{9}{40} + \frac{10}{40} = \frac{19}{40} = 0.475$$
Answer: $\dfrac{19}{40} = 0.475$, or $47.5\%$.
Intuition
This problem is a basic application of the law of total probability combined with independence. The key structural point is that matching can happen in multiple mutually exclusive ways (match on red or match on blue), so you compute each separately and add. Notice the answer is slightly below $1/2$ -- the jars have different compositions, so matching is a bit harder than if they were identical. If both jars had the same color proportions, the match probability would be $p_{\text{red}}^2 + p_{\text{blue}}^2$, which equals $1/2$ only when each color has proportion $1/2$. The more skewed the proportions, the higher the match probability (in the extreme, if one color has proportion $1$, you always match). This is related to the concept of the Herfindahl index in economics -- the probability of a random match measures concentration.