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A tree diagram is a visual representation used to map out and solve probability problems. It shows all possible outcomes of an event and their corresponding probabilities. Here's how to use a tree diagram for this exercise:

Draw a branch for each possible outcome for the first tulip bulb. Since we are dealing with three colors, the first set of branches will be for red, yellow, and purple.

Each of these branches will split into further branches representing the outcome of the second tulip bulb. For example, if the first tulip bulb chosen is red, the next set of branches will represent the possibility of selecting a red, yellow, or purple bulb next.

This way, a tree diagram helps you visualize all possible outcomes and makes it easier to calculate combined probabilities by following the paths through the branches.

Conditional probability is the probability of an event occurring given that another event has already occurred. In the tulip bulb exercise, we use conditional probability to determine the likelihood of specific sequences of events. Here’s how it applies:

When calculating the probability of selecting two red bulbs consecutively, the probability of the second event depends on the first event.

First, calculate the probability of selecting a red bulb: \(\frac{12}{30}\). This is straightforward as there are 12 red bulbs out of 30.

Then, for the second selection, the context has changed because one red bulb has already been picked. Now, there are 11 red bulbs left out of a total of 29 bulbs. So, the probability of picking a red bulb the second time is \(\frac{11}{29}\).

This dependency on previous outcomes exemplifies conditional probability. Similar steps are taken for the combinations involving yellow and red bulbs.

Combinations involve selecting items without regard to the order. In this scenario, combinations are used to calculate the likelihood of picking bulbs in different orders.

For example, the probabilities of picking a red first, then yellow, and yellow first, then red, are calculated separately. However, for the final probability of one red and one yellow bulb in any order, we need to consider both combinations.
Calculate each sequence individually, like so:

• Red first, then yellow: \(\frac{12}{30} \times \frac{10}{29}\).
• Yellow first, then red: \(\frac{10}{30} \times \frac{12}{29}\).

Adding these probabilities:
\(\frac{4}{29} + \frac{4}{29} = \frac{8}{29}\).

This demonstrates the idea of combinations - the order might change, but the overall likelihood of different sequences happening sums up to the total probability of the event occurring.