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In a binomial distribution, skewness helps us understand how the probabilities are spread across possible outcomes. Skewness refers to the asymmetry in the probability distribution. If on drawing a graph, the distribution tails off to the left, it is left-skewed, whereas if it tails off to the right, it is right-skewed. A binomial distribution becomes symmetric when the probability of success, denoted as \( p \), is 0.5. Since we often do not have perfectly balanced probabilities, it’s crucial to know how skewness behaves:

• If \( p = 0.5 \), the distribution is symmetric.
• If \( p > 0.5 \), the skewness is towards the left. Here, the high probability of success means fewer extremely low outcomes.
• If \( p < 0.5 \), the skewness is to the right, due to having fewer high-success outcomes.

For our example where \( n = 10 \) and \( p = 0.85 \), the distribution is skewed to the left. This left skewness indicates that the outcomes cluster around higher success numbers due to the large probability of successes.

The expected value is a reliable measure that predicts the average outcome of a probability distribution. Think of it as the center of mass in the distribution. The expected value for a binomial distribution with \( n \) trials and probability of success \( p \) is given by:

• \( E(X) = n \times p \)

This formula essentially averages out all possible outcomes weighed by their probabilities. For the given problem where \( n = 10 \) and \( p = 0.85 \), we compute the expected value as:

• \( E(X) = 10 \times 0.85 = 8.5 \)

This implies that, on average, about 8.5 out of 10 trials are expected to be successes. The concept of expected value is foundational in probability as it guides us about the most likely long-term result when the action is repeated many times.

Probability analysis helps us estimate the likelihood of certain outcomes in a probability distribution. When determining probabilities like \( P(r \leq 3) \), it’s vital to consider how skewness affects distribution mass. Given that our distribution is skewed left:
- The majority of the probability mass is concentrated towards the higher end of the spectrum.
This makes lower outcomes like \( r \leq 3 \) relatively rare, thus the probability is low. With \( p = 0.85 \), most trials result in successes, so observing three or fewer successes in this case is not very probable. Hence, we expect \( P(r \leq 3) \) to be small, aligning with the overall distribution shape and the calculated expected value of 8.5.

Success probability refers to the likelihood of achieving a certain number of successful outcomes in a given number of trials. For binomial distributions skewed left, such as ours with \( p = 0.85 \), expecting a high number of successes is typical:

• The distribution's left skewness already suggests a concentration of outcomes around higher values.
• Given \( E(X) = 8.5 \), most probability is around this average.
• Thus, events with \( r \geq 8 \) fall around the center or even above where most of the distribution is.

Given these conditions, \( P(r \geq 8) \) is quite high since these outcomes resonate with the cluster of probability mass, making such successful outcomes very likely. Understanding this guides us when anticipating outcomes in scenarios with a high probability of success.