Problem 24 - Algebra 2 and Trigonometry

Question

At a carnival, Chrystal is managing a game in which a dart is thrown at a square board with a bull's-eye in the center. The board measures 3 feet by 3 feet and the bull's-eye has a 1 -inch radius. Players who hit the bull's-eye receive a prize. a. Assume that each player is unskilled and each throw is radom but always lands within the square. What is the theoretical probability that a player will hit the bull's-eye? b. At the end of the day, Chrystal finds that she gave out 72 prizes for \(1,270\) throws. What is the experimental probability that a player hit the bull's-eye? c. Are the theoretical probability and the empirical probability the same? If not, explain why they are different.

Step-by-Step Solution

Answer

Theoretical probability is \( \frac{\pi}{1296} \); experimental probability is \( \frac{72}{1270} \). They differ due to real-world variations.

Step 1: Convert board and bull's-eye measurements

First, we need to convert the measurements to the same unit. The board is 3 feet by 3 feet, which is 36 inches by 36 inches, because 1 foot = 12 inches. The radius of the bull's-eye is already in inches, with a 1-inch radius.

Step 2: Calculate the area of the board

To find the area of the square board, use the formula for the area of a square: \( A = s^2 \), where \( s \) is the side length. So, the area of the board is \( 36 \times 36 = 1296 \) square inches.

Step 3: Calculate the area of the bull's-eye

The area of a circle is found using the formula \( A = \pi r^2 \), where \( r \) is the radius. So, the area of the bull’s-eye is \( \pi \times 1^2 = \pi \) square inches.

Step 4: Calculate theoretical probability

The theoretical probability of hitting the bull's-eye is the ratio of the area of the bull's-eye to the area of the board. So, \( P(\text{hit}) = \frac{\pi}{1296} \).

Step 5: Calculate experimental probability

The experimental probability is the ratio of successful outcomes (prizes awarded) to the total number of trials (throws). Therefore, \( P(\text{hit}) = \frac{72}{1270} \).

Step 6: Compare theoretical and experimental probabilities

The theoretical probability \( \frac{\pi}{1296} \) is approximately \( 0.00242 \), while the experimental probability \( \frac{72}{1270} \) is approximately \( 0.05669 \). These probabilities are different because theoretical probability assumes perfect randomness and no external factors, while experimental results can be affected by errors, player skill level variation, or measurement inaccuracies.

Key Concepts

Understanding Probability Theory

Probability theory is the foundation of determining how likely events are to happen. It helps us evaluate the chance of outcomes in different scenarios, just like Chrystal's dartboard at the carnival. When you calculate the theoretical probability, you are determining the likelihood of an event under ideal circumstances.
For example, theoretical probability focuses on the chance of hitting the bull's-eye, assuming the dart is thrown randomly and lands within the square-shaped board. This setting considers no real-world interference, meaning nobody is exceptionally skilled, and darts are not affected by the environment.

• Theoretical probability is calculated by dividing the favorable outcomes by all possible outcomes.
• In Chrystal’s scenario, we consider the area of the bull's-eye versus the whole board to calculate this probability.

By converting the board's measurements to a consistent unit and comparing the areas, we obtain the probability that seems ideal:
\( P(\text{hit}) = \frac{\pi}{1296} \).
Theoretical probabilities guide us in creating expectations for various situations, helpful for predicting how events should pan out under this perfect randomness.

Defining and Calculating Experimental Probability

Experimental probability comes into play when theoretical assumptions meet actual results.
It uses real-world data to estimate the probability of an event. In Chrystal's carnival dart game, she recorded that 72 prizes were awarded out of 1,270 throws. Experimental probability uses this information:

• The probability is found by dividing the number of successful events (prizes) by the total number of trials (throws).
• In this context, the experimental probability is calculated as \( P(\text{hit}) = \frac{72}{1270} \).

Experimental probabilities often differ from theoretical ones due to a variety of factors like human error, environmental conditions, or players' skill levels.
This real-world approach gives us a pragmatic look at probability by focusing on what actually occurs rather than what should happen under idealized conditions.

Comparison of Probabilities: Theoretical vs. Experimental

Examining the differences between theoretical and experimental probabilities allows us to gain insights into how closely practice matches theory—as well as why discrepancies may arise.
For Chrystal’s dart game, the theoretical probability of hitting the bull's-eye was around 0.00242, while the experimental probability was approximately 0.05669.

• These numbers are evidently different, highlighting that theory does not always align with real-world occurrences.
• Theoretical odds assume perfect randomness without external interference, whereas experimental odds reflect the complex nature of live environments.

A variety of factors can influence these outcomes:

• Human error or lack of skill might lead to more or less frequent hits.
• Environmental conditions, such as wind or board placement, might affect results.

Understanding these differences helps in refining expectations and can guide improvements or adjustments in the setup or execution of similar games or experiments in the future.