Question

Refer to Exercise \(5.66 .\) Twenty people are asked to select a number from 0 to \(9 .\) Eight of them choose a \(4,5,\) or 6 a. If the choice of any one number is as likely as any other, what is the probability of observing eight or more choices of the numbers \(4,5,\) or \(6 ?\) b. What conclusions would you draw from the results of part a?

Short answer

Answer: The probability of at least 8 out of 20 people selecting a number between 4 and 6 is approximately 0.7356 or 73.56%.

Step-by-step solution

Identify the probability of success in a single trial

Since we have 10 possible numbers to choose from (0-9), and 3 of them are 4, 5, or 6, the probability of success in a single trial can be calculated as: \(p = \frac{3}{10} = 0.3\)

Calculate the cumulative probability for observing at least 8 choices

We want to find the probability of observing 8 or more choices of the numbers 4, 5, or 6. To do this, we will calculate the cumulative probability for observing 8 to 20 choices by summing the individual probabilities: \(P(X \geq 8) = \sum_{k=8}^{20} \binom{20}{k}(0.3)^k(1-0.3)^{20-k}\) Now, we'll calculate the binomial probabilities for each value of \(k\) and sum them up: \(P(X \geq 8) \approx 0.7356\)

Interpret the results

The probability of observing eight or more choices of the numbers 4, 5, or 6 is approximately \(0.7356\), which is quite high. This implies that the occurrence of at least 8 out of 20 people picking a number between 4 and 6 is not very surprising or unusual, given the assumption that the choice of any one number is as likely as choosing any other. Therefore, we can conclude that the results align with the assumption of equal likelihood for selecting each number.

Key concepts

Binomial Probability

Binomial probability deals with situations where an event with two possible outcomes, often called "success" and "failure," is repeated multiple times. Each trial is independent, and the probability of success is consistent across all trials. In the context of our exercise, the success is defined as a person selecting one of the numbers 4, 5, or 6. This is out of a total of 10 possible numbers, making the probability of success for a single trial 0.3, as there are 3 favorable outcomes out of 10 possible choices.

• In any binomial probability setting, we ask questions like "What is the probability of achieving exactly k successes in n trials?"
• Or, as in our exercise, "What is the probability of achieving at least a certain number of successes?"

To calculate these, we use the formula:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose k successes in n trials.

Cumulative Probability

Cumulative probability gives us the probability of observing a value within a given range in a probability distribution. In this case, it is about finding the likelihood of observing at least 8 successes out of 20 trials.

• This means summing up the probabilities of having 8, 9, 10, ... up to 20 successes, which are all the possible outcomes greater than or equal to 8.
• We compute it by adding each specific probability from the binomial formula for each success count, from 8 through 20, using the binomial probability formula for each case.

The expression for cumulative probability in this exercise is:\[P(X \geq 8) = \sum_{k=8}^{20} \binom{20}{k} (0.3)^k (0.7)^{20-k}\]By calculating these values, we find a cumulative probability of approximately 0.7356, indicating that such an event is relatively common.

Probability Distribution

A probability distribution presents a comprehensive description of how the probabilities are distributed over the possible outcomes of a random variable. In our case, a binomial distribution is relevant, because it is used when dealing with a fixed number of independent trials, each having two possible outcomes.

• The binomial distribution is determined by two parameters: the number of trials \(n\), and the probability of success \(p\) in each trial.
• For our exercise, the distribution can be visualized as a set of bars, each representing the probability of achieving exactly k successes in 20 trials.

Essential attributes of a binomial distribution are its mean \(\mu\) and variance \(\sigma^2\), which can be calculated as:\[\mu = np\]\[\sigma^2 = np(1-p)\]For instance, here \(\mu = 20 \times 0.3 = 6\), which represents the expected number of successes out of 20 trials.

Statistical Inference

Statistical inference involves making conclusions about populations based on data from samples. It provides a way to draw insights and make decisions in the presence of uncertainty.

• In our situation, we infer whether the choice pattern (8 out of 20 people choosing 4, 5, or 6) aligns with the assumption of equal probability for every digit.
• The high cumulative probability (approximately 0.7356) suggests that this observed event is not unusual. Therefore, we conclude that the results do support the equal likelihood of choosing any digit from 0 to 9.

Statistical inference often involves hypothesis testing, confidence intervals, and estimation. Through these tools, we can objectively assess whether observed data aligns with theoretical models or assumptions.