Cashew Nut Favorite Probability Calculation And Analysis

In the realm of probability and statistics, intriguing scenarios often arise when exploring people's preferences and choices. In this article, we delve into such a scenario, focusing on the favorite nut among adults. Specifically, we address the statement that 32% of adults favor cashews as their top nut choice. We embark on a statistical journey, randomly selecting 12 adults and inquiring about their preferred nut. Our mission is to calculate the probability that a certain number of these individuals will declare cashews as their favorite. We aim to unravel the probabilities associated with two specific outcomes: (a) exactly three individuals naming cashews as their favorite, and (b) at least three individuals sharing this preference.

Let's start by dissecting the first part of our problem: determining the probability that exactly three out of the 12 randomly selected adults will choose cashews as their favorite nut. This scenario falls under the umbrella of binomial probability, a statistical tool employed when dealing with a fixed number of independent trials, each having only two possible outcomes: success or failure. In our case, a "success" is an adult selecting cashews as their favorite, while a "failure" is an adult choosing any other nut.

The bedrock of binomial probability calculations lies in the binomial probability formula. This formula empowers us to compute the probability of obtaining a specific number of successes in a set number of trials. The formula is expressed as follows:

$$P(X = k) = {n extbackslash choose k} * p^k * (1 - p)^{(n - k)}$$

Where:

• P(X = k) signifies the probability of achieving exactly k successes.
• n represents the total number of trials.
• k denotes the number of successes we are interested in.
• p stands for the probability of success in a single trial.
• ${n extbackslash choose k}$ is the binomial coefficient, calculated as n! / (k!(n - k)!), where "!" signifies the factorial function.

Now, let's apply this formula to our specific scenario. We have 12 adults (n = 12), and we want to find the probability that exactly 3 of them favor cashews (k = 3). The probability of an adult preferring cashews is 32%, or 0.32 (p = 0.32). Plugging these values into the binomial probability formula, we get:

$$P(X = 3) = {12 extbackslash choose 3} * (0.32)^3 * (1 - 0.32)^{(12 - 3)}$$

Let's break down this calculation step by step:

1. Calculate the binomial coefficient ${12 extbackslash choose 3}$, which is 12! / (3!9!) = 220.
2. Calculate $(0.32)^3$ ≈ 0.032768.
3. Calculate $(1 - 0.32)^{(12 - 3)}$, which is $(0.68)^9$ ≈ 0.03168.

Now, multiply these values together:

$$P(X = 3) = 220 * 0.032768 * 0.03168 ≈ 0.2284$$

Therefore, the probability that exactly three out of the 12 randomly selected adults will choose cashews as their favorite nut is approximately 0.2284 or 22.84%. This result highlights the likelihood of observing this specific outcome in our sample.

The second part of our exploration takes us into the realm of calculating the probability that at least three adults out of our sample of 12 favor cashews. This entails a slightly more intricate calculation, as we need to consider multiple scenarios. "At least three" implies that three or more adults could prefer cashews, meaning we need to account for the probabilities of 3, 4, 5, 6, 7, 8, 9, 10, 11, or even all 12 adults choosing cashews.

One approach to tackling this problem is to calculate the probabilities for each of these individual scenarios (3, 4, 5, ..., 12) using the binomial probability formula, as we did in part (a), and then sum them up. However, this method can be quite tedious and time-consuming. A more efficient approach involves leveraging the concept of complementary probability.

The complementary probability principle states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring. In our case, the event we're interested in is "at least three adults prefer cashews." The complement of this event is "less than three adults prefer cashews," which means zero, one, or two adults favor cashews.

Therefore, we can calculate the probability of at least three adults preferring cashews by subtracting the probability of less than three adults preferring cashews from 1. Mathematically, this can be expressed as:

$$P(X ≥ 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)]$$

Now, let's calculate the probabilities for P(X = 0), P(X = 1), and P(X = 2) using the binomial probability formula:

• For P(X = 0):

  $$P(X = 0) = {12 extbackslash choose 0} * (0.32)^0 * (0.68)^{12} ≈ 1 * 1 * 0.01129 ≈ 0.01129$$

• For P(X = 1):

  $$P(X = 1) = {12 extbackslash choose 1} * (0.32)^1 * (0.68)^{11} ≈ 12 * 0.32 * 0.0166 ≈ 0.0637$$

• For P(X = 2):

  $$P(X = 2) = {12 extbackslash choose 2} * (0.32)^2 * (0.68)^{10} ≈ 66 * 0.1024 * 0.0244 ≈ 0.1643$$

Now, let's plug these values back into our equation:

$$P(X ≥ 3) = 1 - [0.01129 + 0.0637 + 0.1643] ≈ 1 - 0.23929 ≈ 0.76071$$

Therefore, the probability that at least three out of the 12 randomly selected adults will choose cashews as their favorite nut is approximately 0.76071 or 76.071%. This result underscores the relatively high likelihood of observing at least three cashew enthusiasts in our sample.

In this statistical exploration, we have successfully calculated the probabilities associated with cashew nut preference among adults. We discovered that the probability of exactly three out of 12 adults choosing cashews as their favorite is approximately 22.84%, while the probability of at least three adults sharing this preference is a substantial 76.071%. These findings shed light on the distribution of preferences within our sample and provide valuable insights into the likelihood of observing specific outcomes. The application of binomial probability and complementary probability principles has enabled us to navigate this statistical landscape and arrive at meaningful conclusions. This exercise exemplifies the power of probability in unraveling the complexities of real-world scenarios and provides a framework for analyzing similar preference-based inquiries.