Problem 5 - Algebra 2 and Trigonometry

Question

A spinner is divided into five equal sections numbered 1 through \(5 .\) The arrow is equally likely to land on any section. Find the probability of: a. an odd number on any one spin b. at least three odd numbers on four spins c. at least two odd numbers on four spins d. at least one odd number on four spins

Step-by-Step Solution

Answer

a. \(\frac{3}{5}\) b. \( \frac{387}{625} \) c. \( \frac{541}{625} \) d. \( \frac{649}{625} \)

Step 1: Understanding the Probability of One Spin

The spinner has five equal sections: 1, 2, 3, 4, 5. Odd numbers in these sections are 1, 3, and 5. There are 3 odd numbers out of 5 total sections, so the probability of landing on an odd number in one spin is \( \frac{3}{5} \).

Step 2: Probability of At Least Three Odd Numbers on Four Spins

For four spins, compute the probability of exactly three odd numbers, and the probability of all four being odd. Use the binomial distribution formula: \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \). Here, \( n = 4 \), \( p = \frac{3}{5} \), and \( k = 3 \) or \( k = 4 \). Compute:- Exactly three odd: \( \binom{4}{3} \left(\frac{3}{5}\right)^3 \left(\frac{2}{5}\right)^1 \)- Exactly four odd: \( \binom{4}{4} \left(\frac{3}{5}\right)^4 \)Calculate and add both probabilities to get \( P(X \geq 3) \).

Step 3: Probability of At Least Two Odd Numbers on Four Spins

Use the same binomial formula. Calculate probabilities for exactly two, exactly three, and exactly four odd numbers:- Exactly two odd: \( \binom{4}{2} \left(\frac{3}{5}\right)^2 \left(\frac{2}{5}\right)^2 \)- Exactly three odd and exactly four odd have already been computed in the previous step. Sum these probabilities for \( P(X \geq 2) \).

Step 4: Probability of At Least One Odd Number on Four Spins

Calculate the probability of getting zero odd numbers (none) and subtract from 1. Use binomial formula for none:- Exactly zero odd: \( \binom{4}{0} \left(\frac{2}{5}\right)^4 \)Subtract from 1 to find \( P(X \geq 1) = 1 - P(X = 0) \).

Key Concepts

Probability

Probability is a measure of how likely an event is to occur. It is expressed as a ratio or fraction:

• If an event is certain to happen, its probability is 1.
• If an event cannot happen, its probability is 0.

In the case of our spinner, which has five equal sections numbered 1 to 5, we're interested in finding the probability of landing on an odd number in a single spin. The odd numbers available on the spinner are 1, 3, and 5. Hence, there are 3 favorable outcomes out of 5 possible outcomes.
This makes the probability of landing on an odd number in any one spin equal to:\[ P(\text{odd number}) = \frac{3}{5} \] Understanding basic probability helps in calculating more complex scenarios, such as landing on multiple odd numbers over several spins.

Odd Numbers

Odd numbers are integers that cannot be divided evenly by 2. When a number ends in 1, 3, 5, 7, or 9, it is classified as odd. These numbers are important in our problem, as we want to calculate how frequently the spinner lands on these sections.
On our spinner, the odd numbers are particularly important because understanding how often the spinner lands on them informs us about possible outcomes in scenarios involving multiple spins. Calculating the probability of landing on at least one or more odd-numbered sections over several spins involves recognizing these repetitions and computing their likelihood using probability theory.

Spinner

A spinner is a simple tool used in probability to illustrate random selection among different outcomes. Our spinner is divided into five equal parts, each numbered from 1 to 5.
When the spinner is used, it can land on any of the five sections with equal likelihood. This assumption is crucial as it helps us apply probability rules uniformly across each section. When dealing with this spinner, you only need to count the number of sections with the desired property (e.g., odd numbers) and compare it to the total sections to find a probability.
Thus, understanding the setup of the spinner is integral in solving problems related to its random outcomes, particularly when combined with advanced statistical methods like the binomial distribution.

Binomial Formula

The binomial formula is essential for determining probabilities when dealing with multiple trials or events, like spinning a spinner multiple times. The formula is:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

• \( n \) is the total number of trials (or spins).
• \( k \) is the number of successful outcomes you are counting.
• \( p \) is the probability of one specific successful outcome in a single trial.
• \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) successes out of \( n \) trials.

In probability scenarios with multiple trials, like our spinner being spun four times, the binomial formula provides a way to calculate the likelihood of achieving a desired number of specific outcomes. It accounts for all possible scenarios of achieving exactly that number of successes, making it invaluable in these multi-step probability problems.