"Planting Tulips
A bag of 30 tulip bulbs purchased from a nursery contains 12 red tulip bulbs, 10 yellow tulip bulbs, and 8 purple tulip bulbs. Use a tree diagram like the one in Example 5 to answer the following:


c. What is the probability that the first bulb selected is yellow and the second is red?
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Verified step by step guidance

1. Identify the total number of tulip bulbs in the bag. Since there are 12 red, 10 yellow, and 8 purple bulbs, the total is \$12 + 10 + 8 = 30$ bulbs.
2. Determine the probability of selecting a yellow bulb first. This is the number of yellow bulbs divided by the total number of bulbs, so \(P(\text{first yellow}) = \frac{10}{30}\).
3. After selecting one yellow bulb, update the total number of bulbs remaining. Since one bulb is taken out, the new total is \$30 - 1 = 29$ bulbs.
4. Determine the probability of selecting a red bulb second, given that the first bulb was yellow. The number of red bulbs remains 12, so \(P(\text{second red} \mid \text{first yellow}) = \frac{12}{29}\).
5. Calculate the combined probability of both events happening in sequence by multiplying the probabilities: \(P(\text{first yellow and second red}) = P(\text{first yellow}) \times P(\text{second red} \mid \text{first yellow}) = \frac{10}{30} \times \frac{12}{29}\).

Key Concepts

Probability of Sequential Events

This concept involves calculating the likelihood of multiple events happening in a specific order. When selecting bulbs one after another without replacement, the outcome of the first event affects the second. The overall probability is found by multiplying the probability of the first event by the conditional probability of the second event.

Conditional Probability

Conditional probability measures the chance of an event occurring given that another event has already happened. In this problem, after selecting the first bulb, the total number of bulbs changes, which affects the probability of selecting the second bulb. Understanding this helps adjust probabilities based on updated sample sizes.

Tree Diagram

A tree diagram is a visual tool that maps out all possible outcomes of sequential events and their probabilities. It helps organize and calculate combined probabilities by branching out each event step-by-step. Using a tree diagram clarifies complex probability problems by showing all paths and their associated probabilities.