Problem 26 - Algebra 2

Question

Anita scores well enough at a carnival game that she gets to randomly draw two prizes out a prize bag. There are 6 purple T-shirts, 8 yellow T-shirts, and 5 T-shirts with a picture of a celebrity on them in the bag. Find each probability. \(P(\text { choosing a yellow, then a purple) }\)

Step-by-Step Solution

Answer

The probability is \( \frac{8}{57} \).

Step 1: Identify Total Prizes

Add up the total number of T-shirts in the prize bag. There are 6 purple T-shirts, 8 yellow T-shirts, and 5 celebrity T-shirts. So, the total is:\[ 6 + 8 + 5 = 19 \]

Step 2: Probability of First Yellow T-shirt

Determine the probability that the first T-shirt drawn is yellow. There are 8 yellow T-shirts out of 19 total, so the probability is:\[ P( ext{first yellow}) = \frac{8}{19} \]

Step 3: Probability of Second Purple T-shirt

After drawing the first T-shirt as yellow, there is one less T-shirt in the bag, and it will not affect the number of purple T-shirts. Thus, the total number of T-shirts now is 18. The probability of drawing a purple T-shirt next is:\[ P( ext{second purple | first yellow}) = \frac{6}{18} = \frac{1}{3} \]

Step 4: Calculate Combined Probability

Multiply the probabilities of each event to find the probability of both occurring in sequence:\[ P( ext{yellow then purple}) = \frac{8}{19} \times \frac{1}{3} = \frac{8}{57} \]

Key Concepts

Conditional Probability

Conditional probability is a way of finding the likelihood of an event happening given that another event has already occurred. In our example, we want to find the probability of picking a purple T-shirt after already having drawn a yellow one from the bag. The process can be easily understood using the formula:

• First, identify the probability of the initial event (choosing a yellow T-shirt).
• Next, assess the probability of the subsequent event (drawing a purple T-shirt) given the initial event has occurred.

In the exercise, the first step was to calculate the probability of picking a yellow T-shirt, which was found to be \( \frac{8}{19} \). Then, knowing one yellow T-shirt is already drawn, the total count reduces, and the probability of selecting a purple T-shirt becomes \( \frac{1}{3} \).
This step-by-step process is crucial in differentiating conditional probability from regular probability calculations.

Combinatorics

Combinatorics forms the core of calculating probabilities when dealing with multiple events. It is the branch of mathematics focused on counting, arranging, and the study of combinations of elements. In our problem, combinatorial thinking helps in deducing how choices affect outcomes. For instance:

• The total number of T-shirts initially is calculated by simply adding them all up: purple, yellow, and celebrity T-shirts, giving us 19 total.
• Combinatorics helps us consider how the total number of T-shirts changes after one is selected, updating the context for subsequent probabilities.

Understanding the basics of combinatorics can allow better insights into how probability spaces shift, especially in scenarios where sequential decisions are made, such as in our example where drawing the first result directly influences the setup for the second draw.

Statistics

Statistics involves the collection, analysis, interpretation, and presentation of data. In our exercise, statistical concepts help interpret and predict probabilities. By breaking down details:

• We assess initial values (total of each type of T-shirt) to form a statistical dataset — a key preparation in any statistical analysis.
• Using statistical knowledge, the sequential probability can be calculated to analyze real-life outcomes and answer practical questions, such as the likelihood of drawing specific sequences of T-shirts.

This understanding aids in making informed decisions or predictions based on observed patterns, such as determining the chances of a particular sequence of events (like drawing a yellow T-shirt followed by purple) occurring within a data sample.