Problem 36 - Algebra 2

Question

OPEN ENDED Describe a situation for which the \(P(2 \text { or more) can be found by }\) using a binomial expansion.

Step-by-Step Solution

Answer

The probability is 0.8125 for at least 2 successes in 5 trials.

Step 1: Understanding the Situation

To use a binomial expansion for calculating the probability of 2 or more occurrences, we need a situation where there is a clear 'success' or 'failure', and a fixed number of trials each with the same probability of success. An example could be finding at least two 'heads' in 5 tosses of a fair coin.

Step 2: Define the Parameters

In our example, each coin toss is a trial. The probability of getting 'heads' in one toss (success) is 0.5, and the number of trials is 5. We are interested in finding the probability of getting 2 or more heads.

Step 3: Applying the Binomial Formula

The binomial formula is given by \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]where \(n\) is the number of trials, \(k\) is the number of desired successes, and \(p\) is the probability of success on a single trial. We'll use this to calculate probabilities for k = 0 and k = 1.

Step 4: Calculate \(P(0)")

To calculate the probability of no heads (0 heads), use: \[P(X = 0) = \binom{5}{0} (0.5)^0 (0.5)^5 = 1 \times 1 \times 0.03125 = 0.03125\]

Step 5: Calculate \(P(1)")

To calculate the probability of exactly 1 head, use: \[P(X = 1) = \binom{5}{1} (0.5)^1 (0.5)^4 = 5 \times 0.5 \times 0.0625 = 0.15625\]

Step 6: Using Complementary Probability

The probability of 2 or more heads is the complement of getting 0 or 1 head: \[P(2 ext{ or more}) = 1 - P(0) - P(1) = 1 - 0.03125 - 0.15625 = 0.8125\]

Step 7: Final Result

After calculating, we find that the probability of getting at least 2 heads in 5 coin tosses is 0.8125.

Key Concepts

Understanding Probability

Probability is a mathematical concept that measures the likelihood of an event occurring. In simple terms, it's a way to quantify uncertainty or chance. Imagine you have a familiar situation, like flipping a coin. Each flip is called a trial and has two possible outcomes: heads or tails. Each outcome is equally likely

• The probability of an event is a number between 0 and 1.
• A probability of 0 means the event is impossible.
• A probability of 1 means the event is certain to happen.

For a fair coin, the probability of getting heads on one flip is 0.5, or 50 percent. If you flip the coin multiple times, you can use probability to predict outcomes of these trials collectively.

The Binomial Formula Explained

The binomial formula helps us compute the probability of getting a certain number of successes in a fixed number of independent trials. It's particularly useful when each trial has two possible outcomes: success or failure. The formula is:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]Here's what each symbol means:

• \(n\) is the total number of trials.
• \(k\) is the number of successes we want.
• \(p\) is the probability of success on a single trial.

To use it, you'll calculate probabilities for each possible number of successes individually, then sum those probabilities if needed. The binomial formula provides a handy way to tackle situations where you have repeated, independent trials.

Understanding Complementary Probability

Complementary probability refers to the concept that the probability of an event not occurring is one minus the probability of it occurring. Suppose you're investigating multiple coin tosses. You want to find out the probability of achieving 2 or more heads. Instead of calculating the probability for achieving exactly 2, 3, 4, or 5 heads separately, you can simplify the calculation.

• Calculate the probability of getting 0 heads.
• Calculate the probability of getting 1 head.
• Sum these probabilities.

Then, subtract that sum from 1 to find the complementary probability of getting 2 or more heads. In many cases, calculating complementary probabilities saves time and effort.

Distinguishing Success and Failure

When using the binomial expansion, it's crucial to clearly define what constitutes success and failure in your trials. Let's take the example of tossing a coin. Here, you must decide which outcome is a success. Is it landing on heads or tails? Once you define success, everything else automatically becomes a failure. In our case study involving a fair coin:

• If 'heads' is a success, then 'tails' is a failure.
• The probability of success (getting heads) is always the same for each trial, typically 0.5 for a fair coin.

With a firm grasp of what success and failure are, you can then accurately compute probabilities using the binomial formula. This clarity helps you to align your calculations with the outcome you care about most.