Question

Carnival. A carnival game offers a \(\$ 100\) cash prize for anyone who can break a balloon by throwing a dart at it. It costs \(\$ 5\) to play, and you're willing to spend up to \(\$ 20\) trying to win. You estimate that you have about a \(10 \%\) chance of hitting the balloon on any throw. a) Create a probability model for this carnival game. b) Find the expected number of darts you'll throw. c) Find your expected winnings.

Short answer

You'll throw about 3.44 darts on average, with expected winnings of $17.20.

Step-by-step solution

Understanding the Problem

Let's review the game rules: Each throw costs $5 with a maximum of $20, allowing for up to 4 throws. There is a 10% probability of winning (hitting the balloon) per throw. The prize for hitting the balloon is $100.

Probability Model

The probability of hitting the balloon in one throw is 0.1 (success), and the probability of missing is 0.9 (failure). We need to calculate the probabilities for different outcomes: hitting the balloon on the 1st, 2nd, 3rd, or 4th throw or never hitting at all. - Probability of winning on the 1st throw = 0.1 - Probability of winning on the 2nd throw = 0.9 * 0.1 - Probability of winning on the 3rd throw = 0.9^2 * 0.1 - Probability of winning on the 4th throw = 0.9^3 * 0.1 - Probability of missing all throws = 0.9^4

Expected Number of Darts

Calculate expected number of darts using outcome probabilities:\[E(X) = 1 \cdot 0.1 + 2 \cdot 0.9 \cdot 0.1 + 3 \cdot 0.9^2 \cdot 0.1 + 4 \cdot 0.9^3 \cdot 0.1 + 4 \cdot 0.9^4\]Plug in the values and solve:\[E(X) = 1 \times 0.1 + 2 \times 0.09 + 3 \times 0.081 + 4 \times 0.0729 + 4 \times 0.6561\]\[= 0.1 + 0.18 + 0.243 + 0.2916 + 2.6244\]\[= 3.439\]So, the expected number of darts is approximately 3.44.

Expected Winnings

Calculate expected winnings using probability and price:- Winning = \(100 - (Number of Throws × \)5)- Expected Winnings = \(100 \times ( 0.1 + 0.9 \times 0.1 + 0.9^2 \times 0.1 + 0.9^3 \times 0.1) - ( 1 \times 0.1 + 2 \times 0.9 \times 0.1 + 3 \times 0.9^2 \times 0.1 + 4 \times 0.9^3 \times 0.1 + 4 \times 0.9^4 ) \times 5 \)Solve:- Expected Winnings = \([100 \times 0.3439] - [3.439 \times 5]\)- = \((34.39) - (17.195)\)- = \(17.195Thus, the expected winnings are approximately \)17.20.

Key concepts

Expected Value

The term "expected value" is crucial in understanding the potential outcomes in situations involving chance, as it gives us an average outcome over a long period. In this carnival game scenario, the expected value helps determine how many can throws on average you will need to break the balloon.
The expected value, represented by the symbol \( E(X) \), is calculated by summing the products of each possible event value and its corresponding probability. For example, if the event is hitting the balloon and its payoff is $100, then you have associated probabilities for hitting it in the 1st, 2nd, 3rd, or 4th try, or not hitting it at all.
In this problem, the expected number of darts you will throw is calculated as follows:
\[ E(X) = 1 \cdot 0.1 + 2 \cdot 0.9 \cdot 0.1 + 3 \cdot 0.9^2 \cdot 0.1 + 4 \cdot 0.9^3 \cdot 0.1 + 4 \cdot 0.9^4 \]
This formula sums up the probability-weighted number of attempts needed to win. In simpler terms, it tells us what the average attempt looks like, indicating you would need approximately 3.44 throws on average.

Probability Distribution

Understanding probability distribution is essential when analyzing games of chance such as this carnival game. A probability distribution provides a complete picture of how likely different outcomes are.

In the balloon game example, the probability distribution involves the chances of hitting the balloon on each successive throw or missing it entirely. Here, some possible outcomes and their corresponding probabilities are illustrated as follows:

• Probability of breaking the balloon on your first try: 0.1
• Probability of breaking it on your second try: 0.9 \( \times \) 0.1
• Probability of doing so on your third try: 0.9^2 \( \times \) 0.1
• Probability on your fourth try: 0.9^3 \( \times \) 0.1
• Probability of not hitting at all after four tries: 0.9^4

The beauty of probability distribution is in how it helps you visualize or compute the total likelihood of various occurrences. Knowing these distributions assists in weighing decisions, such as whether playing the game is a worthwhile investment.

Carnival Games

Carnival games, often known as games of chance, are thrilling because of their unpredictability and the possibility of winning enticing rewards. These games are specially designed to keep you guessing while having fun.

Carnival games, including this balloon-prize dart game, typically operate by providing players a chance to win based on probability. They are designed so that the house, or game operator, often has a better chance of profit. However, understanding concepts like expected value and probability distribution can help bettors make more informed decisions.
Such games are built on mathematical models that balance reward-versus-risk. With the dart game, you pay to play, knowing your odds. You must decide if attempting multiple times to hit the target is worth the cost based on the potential reward versus the statistical likelihood of success.
In the end, these games are not just about luck; they embody a rich interplay between math and strategy, and understanding these concepts formulates a rooted understanding of real-world scenarios involving risk and reward.