What are the odds say, if you spin in a game where there are different rarities, and legendary rarity being the hardest to obtain. At (1%). And there are two legendary rarity only. But you spun 150 times and got the same legendary 5 times?

To determine the odds of obtaining the same legendary item 5 times in 150 spins, we need to delve into probability and binomial distributions. Let's break down the problem:

1. **Probability of getting any legendary item (P_legendary):**
   The probability of obtaining a legendary item is given as 1%, or 0.01.

2. **Probability of getting a specific legendary item (P_specific_legendary):**
   Since there are two legendary items, the probability of obtaining one specific legendary item is:
   \[
   P_{\text{specific\_legendary}} = \frac{0.01}{2} = 0.005
   \]

3. **Number of spins (n):**
   The total number of spins is 150.

4. **Number of times the specific legendary item was obtained (k):**
   The specific legendary item was obtained 5 times.

Now, we need to find the probability of obtaining the same legendary item exactly 5 times in 150 spins. This follows a binomial distribution, where the probability \( P \) of exactly \( k \) successes in \( n \) trials is given by:

\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

Where:
- \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \)
- \( p \) is the probability of success on a single trial (in this case, getting the specific legendary item, which is 0.005)
- \( n \) is the number of trials (150)
- \( k \) is the number of successes (5)

Plugging in the values:

\[
P(X = 5) = \binom{150}{5} (0.005)^5 (0.995)^{145}
\]

Let's calculate this step-by-step:

1. **Binomial coefficient \(\binom{150}{5}\):**
   \[
   \binom{150}{5} = \frac{150!}{5!(150-5)!} = \frac{150!}{5! \cdot 145!}
   \]

2. **Probability terms:**
   \[
   (0.005)^5 = 3.125 \times 10^{-13}
   \]
   \[
   (0.995)^{145} \approx 0.472
   \]

3. **Combine all parts:**
   \[
   P(X = 5) = \binom{150}{5} \times 3.125 \times 10^{-13} \times 0.472
   \]

First, let's find \(\binom{150}{5}\):

\[
\binom{150}{5} = \frac{150 \times 149 \times 148 \times 147 \times 146}{5 \times 4 \times 3 \times 2 \times 1} = 10,379,823
\]

Now, multiply everything together:

\[
P(X = 5) = 10,379,823 \times 3.125 \times 10^{-13} \times 0.472 \approx 1.531 \times 10^{-6}
\]

Thus, the probability of obtaining the same legendary item exactly 5 times in 150 spins is approximately \( 1.531 \times 10^{-6} \), or about 0.0001531%.

This probability is extremely small, indicating that getting the same legendary item 5 times in 150 spins is highly unlikely.

Would you like more details on this calculation or have any other questions?

Here are 8 relative questions for further exploration:

1. What is the binomial distribution and how is it used in probability?
2. How do you calculate the binomial coefficient \(\binom{n}{k}\)?
3. What are some common applications of the binomial distribution?
4. How does changing the number of spins affect the probability of obtaining a specific outcome?
5. What is the significance of the probability value obtained in real-world scenarios?
6. How do rare events in games compare to other statistical phenomena?
7. What are some strategies to increase the likelihood of obtaining rare items in games?
8. How does understanding probability help in making informed decisions in games?

**Tip:** In probability, understanding the concept of independent events is crucial, as the outcome of one event does not affect the outcome of another.

Explore the probability calculation of obtaining the same legendary item 5 times in 150 spins. Understand the binomial distribution and key formulas used in probability. Suitable for advanced high school students and anyone interested in probability and gaming.

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