Question

Chris wants to put six plants in a row on his windowsill. He randomly chooses each plant to be an aloe plant, a basil plant, or a violet. What is the probability that either exactly four of the plants are aloe plants or exactly five of the plants are basil plants?

Answer 1

To calculate the probability, we evaluate each scenario independently and add them: the chance for exactly four aloe plants and the chance for exactly five basil plants. We use combinatorial ways to place specific plants and divide by the total number of possible plant arrangements.

Step-by-step explanation:

The probability that either exactly four of the plants are aloe plants or exactly five of the plants are basil plants can be calculated by considering each scenario separately.

First, for exactly four aloe plants, we can choose 4 out of 6 spots for aloe plants in combinatorial ways (denoted as C(6,4)), and then choose any plants (either basil or violet) for the remaining 2 spots, which gives us 22 possibilities. The probability is (C(6,4) * 22) / 36, since there are 3 choices for each of the 6 plants.

Second, for exactly five basil plants, we choose 5 out of 6 spots for basil plants in C(6,5) ways, and then there is only 1 spot left for the remaining plant, which can be aloe or violet, giving us 2 possibilities. The probability is (C(6,5) * 2) / 36.

To find the total probability of either event occurring, we add the two probabilities together. This gives us the final answer.

Answer 2

Probability is the study of calculating the chances or likelihood of an event occurring out of the total number of events. This likelihood is always presented as part of a whole. It could be in fractions or percentages.

There are two techniques to apply to this problem. First, we use the repeated trial equation:

P = n!/r!(n-r)! * p^(n-r) * q^r

This is used to determine the probability that an event will occur exactly 'r' times out of 'n' trials. p is the probability of success, while q is the probability of failure. Together, they add up to 1. In this case, each kind of plant has 1/3 probability. So, p=1/3 and q=2/3.

Second thing to note when dealing with probability problems are 'hint words'. When you are asked to find the probability of event A 'AND' event B happening, then you multiply their probabilities. However, when you are asked to find the probability of event A 'OR' event B happening, then you add their probabilities. Hence, the solution to this problem is:

P = 6!/4!(6-4)! * (1/3)^6-4 * (2/3)^4 + 6!/5!(6-5)! * (1/3)^6-5 * (2/3)^5
P = 0.596 or 59.6%