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Probability calculation is a method used to determine the likelihood of a specific outcome or event. In the context of the exercise, we are interested in finding the probability of certain numbers of adults believing in reincarnation among a set of six adults.
To calculate the probability, we need to understand a few key components:
- **Number of trials (n):** This is the total number of attempts or samples—in this case, the six adults chosen.
- **Probability of success (p):** Defined as the probability of an adult believing in reincarnation, which is given as 0.2 or 20%.
The goal is to find out the probabilities for three different scenarios:
- Exactly five adults who believe in reincarnation.
- All six adults believing in reincarnation.
- At least five adults believing in reincarnation.
Using these parameters and understanding the basics of probability calculation ensures that you can work through steps involving different statistical methods.

The binomial probability formula is key to solving this exercise. It helps us find the probability of a given number of successes (in our case, adults who believe in reincarnation) in a fixed number of trials. Here’s the formula:
\( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \)
Let’s break this down:
- **\( n \): The number of trials (total adults, which is 6).
- **\( k \): The number of successes (adults believing in reincarnation).
- **\( \binom{n}{k} \): The binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \), which gives the number of ways to choose k successes from n trials.
- **\( p^k \): The probability of k successes.
- **\((1-p)^{n-k} \): The probability of the remaining trials being failures.
Here are the detailed calculations for each scenario:
- For exactly five adults believing in reincarnation:
\[ P(X = 5) = \binom{6}{5} (0.2)^5 (0.8)^1 = 6 \times 0.00032 \times 0.8 = 0.001536 \]
- For all six adults believing in reincarnation:
\[ P(X = 6) = \binom{6}{6} (0.2)^6 = 1 \times 0.000064 = 0.000064 \]
- For at least five adults:
\[ P(X \geq 5) = P(X = 5) + P(X = 6) = 0.001536 + 0.000064 = 0.0016 \]

Statistical significance helps us determine whether a result is likely to be due to chance or if it's meaningful in the context of our study. In this case, we need to find out if having five adults who believe in reincarnation out of six is significantly high.
To make this assessment, we compare the probability we calculated to a standard significance level, often set at 0.05. This level is a threshold below which we consider the result statistically significant.
Here’s the step-by-step process:
- We already calculated the probability of five or more adults believing in reincarnation as 0.0016.
- Compare this to 0.05 (the typical significance level).
Since 0.0016 is much smaller than 0.05, we conclude that having five adults who believe in reincarnation out of six is significantly high. This low probability suggests that the result is not likely due to random chance, making it statistically significant.
This helps us understand the implication of our findings, providing deeper insights into the survey results and showing whether they deviate considerably from what might be expected by chance alone.