Problem 209 - Applied Statistics and Probability for Engineers

Question

The probability that concert tickets are available by telephone is \(0.92 .\) For the same event, the probability that tickets are available through a Web site is \(0.95 .\) Assume that these two ways to buy tickets are independent. What is the probability that someone who tries to buy tickets through the Web and by phone will obtain tickets?

Step-by-Step Solution

Answer

The probability is 0.874.

Step 1: Understand Probability and Independence

When probabilities are given as independent, this means the outcome of one event does not affect the outcome of the other. Here, the availability of tickets by telephone and through a website are independent events.

Step 2: Identify Probabilities

The problem provides two probabilities: the probability of getting tickets by telephone, \( P(T) = 0.92 \), and the probability of getting tickets through the Web, \( P(W) = 0.95 \).

Step 3: Calculate Combined Probability

Since the two events are independent, the probability of both happening is the product of their individual probabilities. Thus, \( P(T \, \text{and} \, W) = P(T) \times P(W) = 0.92 \times 0.95 \).

Step 4: Perform the Multiplication

Compute the product: \( 0.92 \times 0.95 = 0.874 \).

Step 5: State the Final Probability

The probability that someone will obtain tickets either by phone or through the Web is 0.874.

Key Concepts

Independent Events

In probability theory, two events are considered independent when the occurrence of one event does not affect the occurrence of the other.

This is an important concept because it allows us to simplify complex probability calculations. Independent events have no influence on each other, much like tossing a coin and rolling a die at the same time. Each has its own possible outcomes that do not interfere with the other.

For example, in the case of buying concert tickets, getting a ticket by phone does not impact the chance of getting a ticket online.
The probabilities can thus be treated separately. It's crucial to verify independence, as it enables specific methods to calculate combined probabilities effectively.

Combined Probability

When dealing with multiple independent events, the next step is often to calculate the combined probability.

Combined probability refers to the likelihood of both events occurring simultaneously.

For independent events, this can be found by multiplying their individual probabilities. This approach stems from the idea that each event's outcome is unaffected by the other's.

• If you have independent events like getting a ticket by phone (\( P(T) = 0.92 \) ) and by web (\( P(W) = 0.95 \) ), the combined probability of obtaining a ticket through both methods is computed as follows.
• The probability is then obtained using: \( P(T \text{ and } W) = P(T) \times P(W) \), which equals \( 0.92 \times 0.95 \).

This simple multiplication gives a clear, direct solution to finding out the probability of both events happening.

Multiplication Rule for Probability

The multiplication rule for probability is a fundamental concept used to find the combined probability of two independent events.

It states that the probability of two independent events \( A \) and \( B \) occurring together is the product of their separate probabilities.

• For independent events like buying tickets by phone and by web, the rule is applied as \( P(A \text{ and } B) = P(A) \times P(B) \).
• This is because the outcome of one event does not change the likelihood of the other.

In the exercise example, the probability of both getting a ticket by phone and by web is calculated using this multiplication rule: \( 0.92 \times 0.95 = 0.874 \).

It provides a straightforward way to approach problems involving multiple independent events, making probability calculations both quick and accurate.