Introduction to the Probability of Landing on Sector 5 after 10 Spins

In this article, we will explore the probability of landing on sector 5 after 10 spins of a spinner with 10 equally likely sectors. This concept is fundamental in understanding probabilistic scenarios in games, gambling, and other real-world situations. We will break down the problem using the complement rule and provide a detailed step-by-step solution.

The Probability of Landing on Sector 5 in One Spin

First, let's consider the probability of landing on sector 5 in a single spin. Since the spinner has 10 sectors, each numbered from 1 to 10, the probability of landing on any specific sector, including 5, is:

P(5) 1 / 10

The Complement Rule and Its Application

The complement rule is a fundamental concept in probability theory. The complement of an event is the event not occurring. Thus, the probability of not landing on sector 5 in a single spin is:

P(not 5) 1 - P(5) 1 - 1 / 10 9 / 10

Calculating the Probability of Not Landing on Sector 5 in 10 Spins

Next, we need to calculate the probability of not landing on sector 5 in 10 consecutive spins. Since each spin is independent of the others, we can raise the probability of not landing on sector 5 in a single spin to the power of 10:

P(not 5 in 10 spins) (9 / 10)^10

Let's compute this value:

(9 / 10)^10 ≈ 0.3487

Complementing the Result to Find the Desired Probability

Now, we use the complement rule to find the probability of landing on sector 5 at least once in 10 spins:

P(at least one 5) 1 - P(not 5 in 10 spins) 1 - 0.3487 ≈ 0.6513

This means that the probability of landing on sector 5 at least once after 10 spins is approximately 0.6513 or 65.13%.

Understanding the Ambiguity in the Question

The question could also introduce ambiguity. It might be asking about the probability of landing on sector 5 after 10 spins that are not 5, or the probability of the first 5 occurring on the 11th spin. Let's explore these scenarios as well:

Scenario 1: The First 5 Occurs on the 11th Spin

If we want to know the probability of the first 5 occurring on the 11th spin, regardless of the outcomes of the first 10 spins, it is simply the probability of landing on sector 5 in one spin:

P(11th spin is 5) 1 / 10 0.10 or 10%

Scenario 2: No 5 in the First 10 Spins, 5 on the 11th Spin

For the second scenario, we need to calculate the probability of not landing on sector 5 in 10 spins and then landing on sector 5 on the 11th spin:

P(no 5 in 10 spins) (9 / 10)^10 ≈ 0.3487

P(5 on 11th spin) 1 / 10 0.10

P(no 5 in 10 spins and 5 on 11th spin) (9 / 10)^10 * 1 / 10 ≈ 0.03487

Scenario 3: Landing on at Least One 5 in 10 Spins

The third scenario involves the complementary probability, which we have already calculated:

P(at least one 5 in 10 spins) 1 - (9 / 10)^10 ≈ 0.6513

Scenario 4: Exactly One 5 in 10 Spins

For the fourth scenario, we need to consider the combinations of landing exactly one 5 and nine non-5s:

P(exactly one 5 in 10 spins) C(10, 1) * (1 / 10)^1 * (9 / 10)^9 ≈ 0.3874

Conclusion

The probability of landing on sector 5 at least once after 10 spins is approximately 65.13%. This article has provided a detailed breakdown of the problem, demonstrating the use of the complement rule in calculating probabilities. Understanding these concepts is valuable in various fields, including statistics, probability theory, and game theory.

Keywords Related to Probability and Spin

Keywords related to this topic include: probability, spin, mathematics, spinner, sectors.