Problem 31 - Algebra 2

Question

If a thumbtack is dropped, the probability of it landing point-up is \(0.3 .\) If 10 tacks are dropped, find each probability. \(P(\text { at least } 8 \text { points up })\)

Step-by-Step Solution

Answer

The probability is approximately 0.0016.

Step 1: Understand the Problem

We are given that the probability of one thumbtack landing point-up is 0.3. We want to find the probability that at least 8 out of 10 thumbtacks land point-up. This problem can be solved using the binomial probability formula.

Step 2: Recognize the Type of Distribution

This is a binomial distribution problem where the number of trials is 10, the number of successes (thumbtacks landing point-up) is a random variable, the probability of success in each trial is 0.3, and we need to find the probability of getting at least 8 successes.

Step 3: Binomial Probability Formula

The binomial probability formula is given by \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \(n\) is the total number of trials, \(k\) is the number of successes, and \(p\) is the probability of success. We will need this formula to calculate \(P(X=8)\), \(P(X=9)\), and \(P(X=10)\).

Step 4: Calculate P(X=8)

Using the binomial formula:\[ P(X=8) = \binom{10}{8} (0.3)^8 (0.7)^2 = 45 \times 0.00006561 \times 0.49 = 0.001455 \]

Step 5: Calculate P(X=9)

Using the binomial formula:\[ P(X=9) = \binom{10}{9} (0.3)^9 (0.7)^1 = 10 \times 0.000019683 \times 0.7 = 0.000137781 \]

Step 6: Calculate P(X=10)

Using the binomial formula:\[ P(X=10) = \binom{10}{10} (0.3)^{10} (0.7)^0 = 1 \times 0.0000059049 \times 1 = 0.0000059049 \]

Step 7: Sum the Probabilities

Add the probabilities calculated in Steps 4, 5, and 6:\[ P(X \geq 8) = P(X=8) + P(X=9) + P(X=10) = 0.001455 + 0.000137781 + 0.0000059049 = 0.0015986859 \]

Step 8: Conclude the Solution

Therefore, the probability that at least 8 thumbtacks land point-up when 10 are dropped is approximately 0.0016.

Key Concepts

Probability of Success

When talking about the probability of success in the context of binomial probability, we're referring to the chance that a single trial will result in a particular desired outcome. In our thumbtack problem, the probability of success is the chance that a thumbtack will land point-up. This probability was given as 0.3 or 30%.

Understanding the probability of success is crucial because it helps set the foundation for solving the problem. It is the value that we use repeatedly in the formula to calculate the probabilities of different numbers of successes. Every trial of dropping a thumbtack has a fixed probability of success, which does not change throughout the trials. This makes problems like these simpler to analyze and solve.

It's important to keep in mind that in a binomial distribution, there are only two possible outcomes for each trial—success or failure. In this scenario, "success" is the thumbtack landing point-up, and "failure" would be landing in any other position.

Binomial Distribution

A binomial distribution describes the number of successes in a fixed number of independent trials of a binary experiment. The outcome of each trial can be classified as a success or a failure with a certain probability. In the thumbtack example, we perform 10 trials (dropping 10 thumbtacks), and we're interested in the number of tacks that land point-up.

Key characteristics of a binomial distribution include:

• A fixed number of trials (e.g., 10 thumbtack drops).
• Only two possible outcomes for each trial: success (point-up) or failure (not point-up).
• The probability of success is consistent across all trials (0.3 per thumbtack drop).
• The trials are independent, meaning the outcome of one trial does not affect another.

Understanding binomial distribution is critical because it tells us how to handle the probabilities of success over multiple trials. It allows us to use formulas and calculations to solve problems systematically, as all trials are uniform and follow the same probability pattern.

Probability Calculation

Probability calculation in the binomial context involves finding the likelihood of various numbers of successes in a fixed number of trials. Calculating probabilities requires a good grasp of the binomial formula, which we'll discuss more in the next section. In the exercise, we had to calculate the probability of getting 8, 9, or 10 thumbtacks landing point-up.

To find these probabilities:

• Identify the number of trials: In our case, 10 thumbtack drops.
• Define the probability of success for each trial: Here, it is 0.3.
• Use the binomial formula to compute the probability for each specific outcome (e.g., exactly 8 successes).
• For probabilities "at least" or "at most" a certain number of successes, calculate the probability for each relevant outcome separately and sum them.

Our task involved calculating and summing the probabilities of getting exactly 8, 9, and 10 successes, yielding a final probability of approximately 0.0016.

Binomial Formula

The binomial formula is the cornerstone of probability calculations for binomial distributions. It gives us a mathematical way to determine the likelihood of a given number of successes in a series of trials. This formula is expressed as:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where:

• \( n \) is the total number of trials.
• \( k \) is the number of successes you are interested in.
• \( p \) is the probability of success on a single trial.
• \( (1-p) \) is the probability of failure on a single trial.
• \( \binom{n}{k} \) is the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).

Using this formula, we computed the probabilities for 8, 9, and 10 successful thumbtack landings by substituting the respective values into the formula. This allowed us to sum these individual probabilities to find the probability of at least 8 tacks landing point-up. The binomial formula is essential in accurately calculating these probabilities and understanding the behavior of a series of trials under a binomial distribution.