91Ó°ÊÓ

Probability calculation involves determining the likelihood of an event occurring. To calculate a probability, you generally divide the number of favorable outcomes by the total number of possible outcomes. In the context of Irena's target practice, calculating the probability for each throw involves understanding that because she moves further back after every throw, the probability of hitting the target decreases exponentially.

For Irena's case, we use her initial probability of hitting the target, which is \( \frac{1}{4} \), and then multiply it by \( \frac{2}{3} \) for each subsequent throw. This decrease follows a geometric progression, making each \( n^{th} \) throw's probability given by the geometric sequence formula: \( P_n = \frac{1}{4} \left( \frac{2}{3} \right)^{n-1} \).

- **Key Points**:

• Understand how probabilities change with consecutive trials and distances.
• Recognize the geometric sequence pattern used to determine subsequent probabilities.

An infinite series involves summing a sequence of numbers indefinitely. In mathematics, particularly in probability and calculus, infinite series are crucial when you want to calculate the entirety of an event where each occurrence has a diminishing probability.

In the problem involving Irena, the challenge is finding the probability that she never hits the target. The infinite series sums up all possible probabilities of each throw missing the target. The formula for the sum of an infinite geometric series is \( S = a_1 \left( \frac{1}{1-r} \right) \), where \( a_1 \) is the first term and \( r \) is the common ratio. In Irena's case, this becomes \( \frac{1}{4} \left( \frac{1}{1-\frac{2}{3}} \right) = \frac{3}{4} \).

- **Wisdom Bits**:

• Understand how to apply series sums to probability problems.
• Recognize the significance of convergence (the series sum reaching a limit).

Complementary probability is a concept that states the probability of an event occurring and not occurring must add up to 1. This is handy when determining the probability of all possible outcomes when one outcome's probability is easier to calculate.

For Irena’s target exercises, calculating the probability of never hitting the target involved first assessing the probability of her hitting it at least once. We found that the sum of all probabilities (probability of at least one hit) is \( \frac{3}{4} \) using the infinite series. Therefore, the complementary probability, which is never hitting the target, was found to be \( 1 - \frac{3}{4} = \frac{1}{4} \).

- **Key Idea**:

• Complementing probabilities simplifies the calculations of coupled events.
• Always ensure sum of complementary probabilities equals 1.