Probability of Hitting the Target

Oct 13, 2025 by miodeya.com 77 views

Introduction

In this article, we will explore the concept of calculating the winning probability in a dart game. The game involves throwing black and white darts in a specific order, and the player's goal is to hit a target in a certain number of rounds. We will use the principles of probability and binomial coefficients to derive a formula for calculating the winning probability.

Game Rules

Suppose we're playing the following dart game:

The player can play up to $T$ rounds.
In each round of the game, the player first throws a black dart, and then a white dart.
Each dart independently hits the target with a probability of $p$ for the black dart and $q$ for the white dart.
The player wins the game if they hit the target in the first $T$ rounds.

Probability of Hitting the Target

Let's start by calculating the probability of hitting the target in a single round. Since the black dart hits the target with a probability of $p$ , and the white dart hits the target with a probability of $q$ , the probability of hitting the target in a single round is $pq$ .

Probability of Winning the Game

Now, let's calculate the probability of winning the game in $T$ rounds. We can use the concept of binomial coefficients to derive a formula for the probability of winning the game.

Binomial Coefficients

A binomial coefficient is a number that represents the number of ways to choose $k$ items from a set of $n$ items, without regard to the order of selection. The binomial coefficient is denoted by $\binom{n}{k}$ and is calculated as:

\binom{n}{k} = \frac{n!}{k!(n-k)!}

Probability of Winning the Game using Binomial Coefficients

Let's denote the probability of winning the game in $T$ rounds as $P(T)$ . We can calculate $P(T)$ using the binomial coefficient as follows:

P(T) = \sum_{k=0}^{T} \binom{T}{k} p^k q^{T-k}

where $p$ is the probability of hitting the target with the black dart, and $q$ is the probability of hitting the target with the white dart.

Derivation of the Formula

To derive the formula for $P(T)$ , we can use the concept of conditional probability. Let's denote the event of hitting the target in the first $k$ rounds as $A_k$ , and the event of hitting the target in the remaining $T-k$ rounds as $B_{T-k}$ . We can calculate the probability of winning the game in $T$ rounds as follows:

P(T) = P(A_0 \cap B_T) + P(A_1 \cap B_{T-1}) + \cdots + P(A_T \cap B_0)

Using the concept of conditional probability, we can rewrite the above equation as:

P(T) = \sum_{k=0}^{T} P(A_k) P(B_{T-k})

Substituting the values of $P(A_k)$ and $P(B_{T-k})$ , we get:

P(T) = \sum_{k=0}^{T} \binom{T}{k} p^k q^{T-k}

Example

Suppose we're playing the dart game with the following parameters:

$T = 5$ rounds
$p = 0.6$ (probability of hitting the target with the black dart)
$q = 0.7$ (probability of hitting the target with the white dart)

Using the formula for $P(T)$ , we can calculate the probability of winning the game in 5 rounds as follows:

P(5) = \sum_{k=0}^{5} \binom{5}{k} (0.6)^k (0.7)^{5-k}

Evaluating the above expression, we get:

P(5) = 0.3281

Conclusion

In this article, we have derived a formula for calculating the winning probability in a dart game using the principles of probability and binomial coefficients. We have shown that the probability of winning the game in $T$ rounds is given by the formula:

P(T) = \sum_{k=0}^{T} \binom{T}{k} p^k q^{T-k}

where $p$ is the probability of hitting the target with the black dart, and $q$ is the probability of hitting the target with the white dart. We have also provided an example of how to use the formula to calculate the probability of winning the game in 5 rounds.

References

[1] "Probability and Statistics" by James E. Gentle
[2] "Binomial Coefficients" by MathWorld

Code

Here is some sample code in Python to calculate the probability of winning the game in $T$ rounds:

import math
def calculate_probability(T, p, q):
probability = 0
for k in range(T + 1):
probability += math.comb(T, k) * (p ** k) * (q ** (T - k))
return probability
T = 5
p = 0.6
q = 0.7
probability = calculate_probability(T, p, q)
print("Probability of winning the game in", T, "rounds:", probability)

Note that this code uses the math.comb function to calculate the binomial coefficient, which is available in Python 3.8 and later. If you are using an earlier version of Python, you can use the following code instead:

import math
def calculate_probability(T, p, q):
probability = 0
for k in range(T + 1):
probability += math.factorial(T) / (math.factorial(k) * math.factorial(T - k)) * (p ** k) * (q ** (T - k))
return probability
**Calculating Winning Probability in a Dart Game: Q&amp;A**
=====================================================

**Introduction**
---------------

In our previous article, we explored the concept of calculating the winning probability in a dart game using the principles of probability and binomial coefficients. We derived a formula for the probability of winning the game in $T$ rounds and provided an example of how to use the formula to calculate the probability of winning the game in 5 rounds.

In this article, we will answer some frequently asked questions about calculating the winning probability in a dart game.

**Q: What is the probability of winning the game in a single round?**
---------------------------------------------------------

A: The probability of winning the game in a single round is the product of the probabilities of hitting the target with the black dart and the white dart, which is $pq$.

**Q: How do I calculate the probability of winning the game in $T$ rounds?**
-------------------------------------------------------------------------

A: To calculate the probability of winning the game in $T$ rounds, you can use the formula:

$$P(T) = \sum_{k=0}^{T} \binom{T}{k} p^k q^{T-k}$$

where $p$ is the probability of hitting the target with the black dart, and $q$ is the probability of hitting the target with the white dart.

**Q: What is the binomial coefficient, and how is it used in the formula?**
-------------------------------------------------------------------------

A: The binomial coefficient is a number that represents the number of ways to choose $k$ items from a set of $n$ items, without regard to the order of selection. It is denoted by $\binom{n}{k}$ and is calculated as:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

In the formula for $P(T)$, the binomial coefficient is used to calculate the number of ways to choose $k$ rounds out of $T$ rounds, where $k$ is the number of rounds in which the player hits the target.

**Q: How do I calculate the probability of winning the game in a specific number of rounds?**
-----------------------------------------------------------------------------------------

A: To calculate the probability of winning the game in a specific number of rounds, you can use the formula for $P(T)$ and substitute the desired number of rounds for $T$. For example, to calculate the probability of winning the game in 5 rounds, you can use the formula:

$$P(5) = \sum_{k=0}^{5} \binom{5}{k} p^k q^{5-k}$$

**Q: What is the significance of the probabilities $p$ and $q$ in the formula?**
--------------------------------------------------------------------------------

A: The probabilities $p$ and $q$ represent the likelihood of hitting the target with the black dart and the white dart, respectively. The value of $p$ is the probability of hitting the target with the black dart, and the value of $q$ is the probability of hitting the target with the white dart.

**Q: Can I use the formula to calculate the probability of winning the game in a non-integer number of rounds?**
-----------------------------------------------------------------------------------------

A: No, the formula is only valid for integer values of $T$. If you want to calculate the probability of winning the game in a non-integer number of rounds, you will need to use a different approach, such as using a continuous probability distribution.

**Q: How do I implement the formula in a programming language?**
----------------------------------------------------------------

A: You can implement the formula in a programming language such as Python or MATLAB using the following code:
```python
import math

def calculate_probability(T, p, q):
    probability = 0
    for k in range(T + 1):
        probability += math.comb(T, k) * (p ** k) * (q ** (T - k))
    return probability

# Example usage
T = 5
p = 0.6
q = 0.7
probability = calculate_probability(T, p, q)
print(&quot;Probability of winning the game in&quot;, T, &quot;rounds:&quot;, probability)
</code></pre>
<p>Note that this code uses the <code>math.comb</code> function to calculate the binomial coefficient, which is available in Python 3.8 and later. If you are using an earlier version of Python, you can use the following code instead:</p>
<pre><code class="hljs">import math

def calculate_probability(T, p, q):
    probability = 0
    for k in range(T + 1):
        probability += math.factorial(T) / (math.factorial(k) * math.factorial(T - k)) * (p ** k) * (q ** (T - k))
    return probability
</code></pre>
<h2><strong>Conclusion</strong></h2>
<p>In this article, we have answered some frequently asked questions about calculating the winning probability in a dart game. We have provided a formula for the probability of winning the game in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> rounds and explained how to use the formula to calculate the probability of winning the game in a specific number of rounds. We have also provided an example of how to implement the formula in a programming language.</p>


                
                    
                        

                    
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