Problem 17 - Algebra 2

Question

Marketing A fruit company guarantees that 90\(\%\) of the pineapples it ships will be ripe within four days. Find each probability for a case containing 12 pineapples. No more than 9 are ripe within four days.

Step-by-Step Solution

Answer

The probability that no more than 9 out of 12 pineapples are ripe within four days can be found by summing up individual probabilities for having 0 to 9 successes (ripe pineapples) in 12 trials, with the chance of success in a single trial being 90\(\%\). The detailed calculation involves the use of the binomial probability formula.

Step 1: Understanding the parameters

The first step is to identify the parameters of the binomial distribution. We have n (number of trials) equal to 12, since we have 12 pineapples. p (probability of success in one trial) is 0.9, since each pineapple has a 90\(\%\) chance to be ripe within four days.

Step 2: Calculating the probabilities

We need to calculate the probability for 0, 1, ..., 9 ripe pineapples. The binomial probability formula is used: \(P(k; n, p) = C(n, k) \cdot (p^k) \cdot ((1-p)^{n-k})\), where P(k; n, p) is the probability of having \(k\) successes in \(n\) trials, \(C(n, k)\) is the combination of \(n\) items taken \(k\) at a time. Repeat the calculation for \(k\) from 0 to 9.

Step 3: Summing the probabilities

To get the final answer, sum up all the calculated probabilities from the previous step. This gives the total probability of getting no more than 9 pineapples ripe within four days.

Key Concepts

Probability Calculation

Probability calculation is a fundamental concept in statistics that helps us understand the likelihood of different outcomes. In the context of a binomial distribution, we deal with situations where there are only two possible results: success or failure. To calculate probability, we use specific formulas and methods.

When you have a binomial scenario, like the pineapples ripening within a certain time frame, the binomial probability formula is key. This formula is:

• \(P(k; n, p) = C(n, k) \cdot (p^k) \cdot ((1-p)^{n-k})\)

Here, \(C(n, k)\) represents combinatorics, \(p\) is the probability of success in a single trial, and \(k\) is the number of successes you are examining. For this specific problem, you need to examine the probability for each scenario from 0 to 9 ripe pineapples and then sum up these probabilities to find the desired likelihood. This approach helps in understanding how likely it is to get a certain number of successes in fixed trials.

Combinatorics

Combinatorics is a branch of mathematics that plays a crucial role in probability calculations, especially in the context of binomial distributions. It involves counting and arranging events in ways that help us understand the number of possible outcomes. In the formula for binomial probability, combinatorics is expressed with the notation \(C(n, k)\).

\(C(n, k)\) indicates the number of combinations of \(n\) items taken \(k\) at a time, also known as "n choose k." The formula for this is:

• \[C(n, k) = \frac{n!}{k! (n-k)!}\]

This helps us determine how many ways we can succeed in \(k\) trials out of a total of \(n\) trials. For example, in the pineapple case, if you want to calculate how likely it is to have exactly 9 ripe fruits, you would compute \(C(12, 9)\).

Combinatorics simplifies dealing with large numbers of potential arrangements, offering a structured approach to calculating probabilities.

Success Probability

Success probability is an essential part of the binomial distribution framework, referring to the likelihood that any single trial will result in success. In our pineapple scenario, this probability is represented as \(p\), with a value of 0.9, meaning each individual pineapple has a 90\(\%\) chance of being ripe.

Understanding success probability is crucial because it directly impacts the overall probability calculation for a series of trials. In a binomial distribution, every trial is independent, and the probability of success remains constant across each trial. This constant success probability helps simplify calculations as we can apply the same value \(p\) across all attempts.

In practice, assessing the success probability provides the baseline for calculating how likely it is to achieve a specific outcome in repeated trials, such as determining the chance that a set number of the fruits will ripen in a given timeframe. Keeping this probability consistent allows for easy application of the binomial distribution formulas to real-life situations.