You've decided you want a plant for your room. At the gardening store, there are $4$ different kinds of plants (tulip, fern, cactus, and ficus) and $4$ different kinds of pots to hold the plants (clay pot, plastic pot, metal pot, and wood pot). $\newline$ If you randomly pick the plant and the pot, what is the probability that you won't get a clay pot or a cactus?

Full solution

Q. You've decided you want a plant for your room. At the gardening store, there are

4

different kinds of plants (tulip, fern, cactus, and ficus) and

4

different kinds of pots to hold the plants (clay pot, plastic pot, metal pot, and wood pot).

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If you randomly pick the plant and the pot, what is the probability that you won't get a clay pot or a cactus?

Plant-Pot Combinations: There are $4$ kinds of plants and $4$ kinds of pots, making a total of $4 \times 4 = 16$ different combinations.
Combinations without Clay Pot: To not get a clay pot, there are $3$ other pots to choose from for each plant, so $4$ plants $\times 3$ other pots $= 12$ combinations without a clay pot.
Combinations without Cactus: To not get a cactus, there are $3$ other plants to choose from for each pot, so $4$ pots $\times 3$ other plants $= 12$ combinations without a cactus.
Subtracting Overlap: We counted the combinations without a clay pot and without a cactus separately, but we need to subtract the overlap, which is the combinations without a clay pot and a cactus. There are $3$ other pots and $3$ other plants, so $3 \times 3 = 9$ combinations that are neither a clay pot nor a cactus.
Final Calculation: Now, we add the combinations without a clay pot and without a cactus, then subtract the overlap: $12 + 12 - 9 = 15$ . This is incorrect because we double-counted the combinations without a clay pot and without a cactus. We should have only subtracted the overlap once.