Questions: Based upon the pattern shown, what is the probability that 3 randomly selected individuals all share the same birthday? Assume that you're not one of the three people in the room. Use the following website, https://wwwwolframalpha,com/2 you find the probability. Answer in scientific notation. 2.05647 × 10^-8 0.0000075 0.01271

Based upon the pattern shown, what is the probability that 3 randomly selected individuals all share the same birthday? Assume that you're not one of the three people in the room.

Use the following website, https://wwwwolframalpha,com/2 you find the probability. Answer in scientific notation.
2.05647 × 10^-8
0.0000075
0.01271
Transcript text: Based upon the pattern shown, what is the probability that 3 randomly selected individuals all share the same birthday? Assume that you're not one of the three people in the room. Use the following website, https://wwwwolframalpha,com/2 you find the probability. Answer in scientific notation. $2.05647 \times 10^{\wedge}-8$ 0.0000075 0.01271
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Solution

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Solution Steps

Step 1: Define the Problem

We want to find the probability that 3 randomly selected individuals all share the same birthday. We assume that there are 365 possible birthdays, and we denote the probability of success (all three individuals sharing the same birthday) as \( p = \frac{1}{365} \). The probability of failure (not sharing the same birthday) is \( q = \frac{364}{365} \).

Step 2: Apply the Binomial Probability Formula

The probability of exactly \( x \) successes in \( n \) trials can be calculated using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

For our case, we have:

  • \( n = 3 \) (the number of individuals),
  • \( x = 3 \) (the number of individuals sharing the same birthday),
  • \( p = \frac{1}{365} \),
  • \( q = \frac{364}{365} \).

Substituting these values into the formula gives:

\[ P(X = 3) = \binom{3}{3} \cdot \left(\frac{1}{365}\right)^3 \cdot \left(\frac{364}{365}\right)^{0} \]

Since \( \binom{3}{3} = 1 \) and \( \left(\frac{364}{365}\right)^{0} = 1 \), we simplify this to:

\[ P(X = 3) = 1 \cdot \left(\frac{1}{365}\right)^3 \cdot 1 = \left(\frac{1}{365}\right)^3 \]

Step 3: Calculate the Probability

Calculating \( \left(\frac{1}{365}\right)^3 \):

\[ P(X = 3) = \frac{1}{365^3} \approx 2.06 \times 10^{-8} \]

Final Answer

The probability that 3 randomly selected individuals all share the same birthday is approximately \( 2.06 \times 10^{-8} \).

Thus, the final answer is:

\[ \boxed{2.06 \times 10^{-8}} \]

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