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In probability theory, a probability distribution describes how a random variable behaves. For this exercise, we use the **Binomial Distribution** as the addresses either are found or are not, with a fixed number of trials. Each trial is an independent attempt to find an address, giving us a binary outcome. The probability of success (finding an address) is denoted as 0.7, and the probability of failure is 0.3 (since failing is 1 minus the success rate).
To calculate the binomial probabilities, we use the formula for each possible outcome where \( r \) ranges from 0 to 6:
\[ P(r) = \binom{n}{r} p^r (1-p)^{n-r} \]
where \( n \) is the total number of friends (6), and \( p \) is the probability of success (0.7). Therefore, you calculate these probabilities to understand how likely you are to find addresses for varying numbers of friends. This distribution helps visualize potential outcomes for situations involving random variables like this.

The **mean** of a probability distribution provides the expected value, which is the central tendency or average number of successes. For a binomial distribution, the mean \( \mu \) is calculated by multiplying the number of trials \( n \) by the probability of success \( p \):
\[ \mu = np \]
In this case, with 6 friends and a 70% success rate, you expect 4.2 addresses found.

The **standard deviation** measures the spread of the probability distribution, indicating how much variation exists from the average:
\[ \sigma = \sqrt{np(1-p)} \]
For our situation, calculate it as the square root of \( 6 \times 0.7 \times 0.3 \). The standard deviation reflects uncertainty in the number of successful address findings.
Both parameters, mean and standard deviation, offer insight into the performance of Old Friends service, framing how typically successful the address retrieval will be, and how much the results might differ each time.

A histogram is a graphical representation of data used to summarize the distribution of a data set. For a binomial distribution, a histogram visually illustrates the probabilities of obtaining each possible number of successes.

To create the histogram for this particular problem, plot the number of friends \( r \) found on the x-axis, ranging from 0 to 6, and the calculated probability \( P(r) \) on the y-axis for each corresponding value. Each bar in the histogram represents the probability of finding an address for a certain number of friends.

• Each bar's height is proportional to the probability \( P(r) \), showing how likely you are to find addresses for exactly 0, 1, 2,..., up to 6 friends.
• The area under the histogram represents the total probability, which is always equal to 1.
• This visual aid helps in understanding how likely each outcome is and identifying the most probable number of friends you can expect to get addresses for.

The histogram serves as a practical tool to visualize and comprehend the distribution of probabilities in binomial events like this one.

**Cumulative Probability** provides the likelihood of obtaining a number of successes that is less than or equal to a particular value. It's calculated by summing the individual probabilities of all outcomes up to that number. This concept is crucial in situations where we need to determine the probability of achieving at least a specific number of successes.
In this exercise, to find how many names need to be submitted to be 97% certain of finding at least two addresses, you would use the cumulative function:
\[ P(\text{at least 2}) = 1 - P(0) - P(1) \]
This equation uses the complement rule which says the probability of finding at least two addresses is one minus the probabilities of finding none or only one. You would adjust the number of friends \( n \) being submitted to ensure this cumulative probability surpasses 0.97.

• Calculate the cumulative probability for different \( n \) values.
• Determine the minimum \( n \) where this probability reaches or exceeds 97%.
• This ensures a strategic approach to finding a desired number of addresses with high confidence.

Cumulative probability provides a comprehensive view, allowing decisions based on confident estimates of success rates.