Problem 15 - 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. All 12 are ripe within four days.

Step-by-Step Solution

Answer

The probability that all 12 are ripe within four days is approximately 0.282 or 28.2\%. This means there's about a 28.2\% chance that all 12 pineapples in a case will be ripe within the 4-day period.

Step 1: Identify the variables

In this problem, \( n = 12 \) is the number of trials (i.e., the number of pineapples in a case), \( x = 12 \) is the number of successful outcomes we want (all the 12 pineapples ripe within four days) and \( p = 0.9 \) is the probability of success on each trial (i.e.. the probability that a pineapple will be ripe within four days).

Step 2: Apply the Binomial Probability Formula

The binomial probability formula is as follows: \(P(x; n, p) = \binom{n}{x} \times (p)^x \times (1-p)^{n-x}\). Here, \(\binom{n}{x}\) represents the number of combinations of n items taken x at a time. It's calculated as \( \binom{n}{x}=\frac{n!}{x!(n-x)!} \) where '!' is the factorial operator. In this case, the binomial coefficient \(\binom{12}{12}\) will be 1, given that there's only one way to choose all 12 pineapples out of 12.

Step 3: Substitute the variables into the formula and simplify

Plugging in the values into the binomial probability formula gives: \( P(12; 12, 0.9) = \binom{12}{12} \times (0.9)^{12} \times (1-0.9)^{12-12} = 1 \times (0.9)^{12} \times (0.1)^0 = 0.9^{12} = 0.282429536481 \).

Key Concepts

Binomial Coefficient

The binomial coefficient, often denoted by \( \binom{n}{x} \), is a crucial part of binomial probability calculations. It represents the number of ways to choose \( x \) successes (or specific events) in \( n \) trials or items. This number can be calculated using the formula:

• \( \binom{n}{x} = \frac{n!}{x!(n-x)!} \)

This formula counts the possible combinations of outcomes without caring about the order in which they occur.

In the context of our problem, where we want all 12 pineapples to be ripe out of a batch of 12, the binomial coefficient \( \binom{12}{12} \) sells as 1. This is because there is exactly one way to choose all 12 pineapples from a group of 12.

Understanding this concept is essential when you need to calculate the likelihood of any number of successes in a set number of trials, such as determining probabilities in quality assurance or predicting outcomes from various tests.

Probability of Success

The probability of success, denoted by \( p \) in binomial statistics, is the likelihood that any individual trial will succeed. For instance, if you are shooting basketballs and you only succeed 60% of the time, then \( p = 0.6 \).

Within the scope of our exercise, \( p = 0.9 \) is the probability that a single pineapple will be ripe within four days. In binomial terms, ripening is your "success."

When you want to know the probability of having x successes out of n trials, the value of \( p^x \) helps build the picture. You multiply the success probability for each successful trial, and that is why power functions \((p)^x\) appear in the formula. This multiplication reflects the likelihood of successive events all turning out favorably.

Thus, the probability of success is a cornerstone of understanding real-life situations that function as binomial processes, such as determining the chance that a new batch of technology will work or estimating if enough workers show up at the office the following day.

Factorial Operator

The factorial operator, symbolized by an exclamation mark (!), is a mathematical operation that multiplies a series of descending natural numbers. For any positive integer \( n \), the factorial is the product of all positive integers less than or equal to \( n \). It is denoted as \( n! \).

So, for example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow rapidly with larger numbers and are normally used in permutations and combinations calculations.

In the binomial formula \( \binom{n}{x} = \frac{n!}{x!(n-x)!} \), the factorial helps decide how many different ways x successes can be assigned among n trials. This is why understanding how to compute factorials is essential for computing binomial coefficients and consequently assessing probabilities.

Despite their intimidating notation, factorials involve basic multiplication, making them accessible once the initial hurdle of comprehension is cleared.