Problem 20 - Algebra 2

Question

As a maintenance manager, Jackie Thomas is responsible for managing the maintenance of an office building. When entering a room after hours, the probability that she selects the correct key on the first try is \(\frac{1}{5} .\) If she enters 6 rooms in an evening, find each probability. \(P(\text { always the correct key })\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{15625} \).

Step 1: Understand the Problem

Jackie Thomas has a probability of \( \frac{1}{5} \) to select the correct key for a room on the first try. She needs to enter 6 rooms. We want to find the probability that she selects the correct key on the first try for all 6 rooms.

Step 2: Identify the Probability for One Room

The probability of selecting the correct key on the first try for one room is given as \( \frac{1}{5} \).

Step 3: Compute the Overall Probability for 6 Rooms

Since she needs the correct key for all 6 rooms, we need to multiply the probability for each independent event (selecting the correct key). The probability for all 6 rooms is:\[ \left( \frac{1}{5} \right)^6 \]

Step 4: Simplify the Probability Expression

Calculate the probability:\[ \left( \frac{1}{5} \right)^6 = \frac{1}{15625} \]

Step 5: Concluding the Solution

Thus, the probability that Jackie always selects the correct key for each of the 6 rooms is \( \frac{1}{15625} \).

Key Concepts

Independent Events

In probability theory, an event is often considered independent if the outcome or occurrence of one event does not affect the outcome or occurrence of another. In simpler terms, two events are independent if the probability of one event occurring remains the same, irrespective of the occurrence of the other event.

For instance, when Jackie chooses a key for one room, and then a key for another room, each selection is made without regard to any other. This means each choice remains uninfluenced by her previous attempts. In our exercise, the selection of a key for each room is viewed as an independent event because choosing the correct key for one specific room does not change the likelihood of selecting the correct key in the following room.

Probability Calculation

Calculating probability is the method of determining how likely an event is to occur. The probability of an event is a number between 0 and 1, where 0 signifies impossibility and 1 represents certainty. Typically, the probability of an event is expressed as a fraction, percentage, or ratio.

• When Jackie tries to select the key for one room, the probability is expressed as a fraction, \( \frac{1}{5} \), representing one successful attempt out of five possible tries.
• To find her probability of doing this successfully for several rooms, and particularly in all six rooms during one night, these probabilities are combined into a cumulative calculation using appropriate mathematical rules.

Multiplication Rule

The multiplication rule in probability is a fundamental principle that helps us determine the combined probability of two or more independent events happening simultaneously. When events are independent, the probability of both events occurring is found by multiplying their individual probabilities.

In the context of the exercise, Jackie needs the correct key for each of the 6 rooms. Since each selection is independent, we multiply the probability of success for one room by itself for each of the 6 rooms: \[ \left(\frac{1}{5}\right)^6 \].

This approach is powerful because it models real-life scenarios involving sequences of independent occurrences and helps provide a structured way to handle cumulative probabilities.

Mathematical Expression Simplification

When dealing with probability, especially with multiple events, expressions can quickly become complex. Simplification is essential for making these expressions more comprehensible and easier to interpret.

• The original problem simplifies the computation of the probability of successful attempts over six rooms. By breaking down complex expressions into simpler forms, such as \( \left(\frac{1}{5}\right)^6 = \frac{1}{15625} \), it becomes easier to grasp the exceedingly low likelihood of Jackie succeeding in using the correct key each time without error.
• This simplification process allows a clearer understanding of probability, especially when presented in its simplest fractional form. It ensures students fully appreciate how minor the probability is in practice.