Probability Of Cashew Lovers Statistical Analysis

Jul 9, 2025 by ADMIN 50 views

Probability of Cashew Lovers A Statistical Analysis of Nut Preferences

In the realm of statistical probability, we often encounter scenarios where we need to determine the likelihood of specific outcomes in a series of independent trials. One such scenario arises when we consider the preferences of individuals within a population. In this article, we delve into a fascinating statistical problem involving nut preferences among adults. Specifically, we explore the probability of a certain number of adults choosing cashews as their favorite nut when a random sample is selected. This analysis provides valuable insights into the application of binomial probability, a fundamental concept in statistics, and its relevance in real-world scenarios. We will dissect the problem, step-by-step, providing a clear and concise explanation of the methodologies used to solve it. By the end of this exploration, readers will have a solid understanding of how to calculate probabilities in situations involving binomial distributions, enhancing their statistical literacy and problem-solving skills.

Problem Statement

Imagine that a survey reveals that 32% of adults have a particular fondness for cashews, considering them their favorite among all nuts. Now, picture this: we decide to randomly select 12 adults and pose the simple question: "Which nut do you favor the most?" Our mission is to uncover the probability that, within this group of 12, a specific number of individuals will declare cashews as their ultimate nut choice. More specifically, we are interested in two distinct scenarios:

(a) What is the likelihood that precisely three of these 12 adults will express their preference for cashews?

(b) What is the probability that at least three adults in the selected group will name cashews as their favorite?

This problem exemplifies a classic application of binomial probability, where we deal with a fixed number of independent trials (in this case, selecting 12 adults), each having only two possible outcomes (an adult either favors cashews or does not). The probability of success (an adult favoring cashews) remains constant across all trials. To solve this, we will employ the binomial probability formula and its cumulative counterpart, providing a comprehensive statistical analysis of nut preferences.

(a) Probability of Exactly Three Cashew Lovers

To determine the probability that exactly three out of the 12 randomly selected adults will say cashews are their favorite nut, we employ the binomial probability formula. This formula is a cornerstone of probability theory and is particularly useful in scenarios where we have a fixed number of independent trials, each with two possible outcomes: success or failure. In our context, a "success" is an adult naming cashews as their favorite, and a "failure" is an adult choosing any other nut. The binomial probability formula is expressed as:

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

Where:

P(X = k) is the probability of observing exactly k successes in n trials.
(n choose k) represents the number of combinations of n items taken k at a time, also known as the binomial coefficient. It can be calculated as n! / (k! * (n - k)!), where "!" denotes the factorial function.
p is the probability of success on a single trial (in our case, the probability that an adult favors cashews, which is 0.32).
(1 - p) is the probability of failure on a single trial (the probability that an adult does not favor cashews).
n is the number of trials (the number of adults selected, which is 12).
k is the number of successes we are interested in (in this case, exactly 3 adults).

Now, let's apply this formula to our specific scenario. We have n = 12, k = 3, and p = 0.32. Plugging these values into the formula, we get:

P(X = 3) = (12 choose 3) * (0.32)^3 * (1 - 0.32)^(12 - 3)

First, we calculate the binomial coefficient (12 choose 3):

(12 choose 3) = 12! / (3! * 9!) = (12 * 11 * 10) / (3 * 2 * 1) = 220

Next, we calculate the probability of 3 successes and 9 failures:

(0.32)^3 ≈ 0.032768

(1 - 0.32)^(12 - 3) = (0.68)^9 ≈ 0.037798

Finally, we multiply these values together:

P(X = 3) = 220 * 0.032768 * 0.037798 ≈ 0.2724

Therefore, the probability that exactly three out of the 12 randomly selected adults will say cashews are their favorite nut is approximately 0.2724 or 27.24%. This calculation underscores the power of the binomial probability formula in quantifying the likelihood of specific outcomes in situations involving repeated independent trials.

(b) Probability of At Least Three Cashew Lovers

Next, we want to find the probability that at least three of the 12 selected adults favor cashews. This means we need to calculate the probability of having 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12 adults who prefer cashews. A straightforward approach would be to calculate each of these probabilities individually using the binomial probability formula, as we did in part (a), and then sum them up. However, there's a more efficient method using the concept of complementary probability.

The probability of "at least three" successes is the complement of the probability of "less than three" successes. In other words:

P(X ≥ 3) = 1 - P(X < 3)

This is because the total probability of all possible outcomes must equal 1. The event "less than three" successes includes the cases where we have 0, 1, or 2 adults favoring cashews. Therefore, we can calculate the probability of at least three successes by subtracting the probabilities of 0, 1, and 2 successes from 1.

P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)

Let's calculate these probabilities individually using the binomial probability formula:

P(X = 0) = (12 choose 0) * (0.32)^0 * (0.68)^12
P(X = 1) = (12 choose 1) * (0.32)^1 * (0.68)^11
P(X = 2) = (12 choose 2) * (0.32)^2 * (0.68)^10

Calculating these values:

(12 choose 0) = 1
(0.32)^0 = 1
(0.68)^12 ≈ 0.009059

P(X = 0) ≈ 1 * 1 * 0.009059 ≈ 0.009059

(12 choose 1) = 12
(0.32)^1 = 0.32
(0.68)^11 ≈ 0.013322

P(X = 1) ≈ 12 * 0.32 * 0.013322 ≈ 0.051188

(12 choose 2) = (12 * 11) / (2 * 1) = 66
(0.32)^2 = 0.1024
(0.68)^10 ≈ 0.019591

P(X = 2) ≈ 66 * 0.1024 * 0.019591 ≈ 0.132148

Now, we sum these probabilities to find P(X < 3):

P(X < 3) ≈ 0.009059 + 0.051188 + 0.132148 ≈ 0.192395

Finally, we subtract this from 1 to find P(X ≥ 3):

P(X ≥ 3) = 1 - P(X < 3) ≈ 1 - 0.192395 ≈ 0.807605

Therefore, the probability that at least three of the 12 randomly selected adults will say cashews are their favorite nut is approximately 0.8076 or 80.76%. This result demonstrates that there's a high likelihood of finding at least three cashew enthusiasts in a group of 12 adults, given the initial statistic that 32% of adults favor cashews.

In this statistical exploration, we've successfully navigated the probabilities associated with nut preferences among adults. We started with the premise that 32% of adults favor cashews and then delved into calculating the likelihood of specific outcomes when selecting a random sample of 12 adults. Through the application of the binomial probability formula and the concept of complementary probability, we determined two key probabilities:

The probability that exactly three out of the 12 adults would name cashews as their favorite nut, which we found to be approximately 27.24%.
The probability that at least three adults in the sample would express their love for cashews, calculated to be around 80.76%.

These results highlight the practical application of binomial probability in real-world scenarios. By understanding and utilizing this statistical tool, we can make informed predictions about the distribution of preferences within a population. The problem-solving approach we've demonstrated here can be extended to a wide range of situations, from market research and opinion polls to quality control and scientific experiments. The ability to quantify uncertainty and make probabilistic assessments is a valuable skill in various fields, and this analysis of nut preferences serves as a clear example of its power and relevance. Furthermore, the use of complementary probability in calculating the "at least three" scenario showcases the importance of strategic thinking in problem-solving, often leading to more efficient and elegant solutions. As we conclude this exploration, we hope readers have gained a deeper appreciation for the role of probability in understanding the world around us and the analytical tools that empower us to make sense of data and uncertainty.