Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the probability of the given event. The coin lands heads all five times.

Step-by-Step Solution

Answer

The probability of the coin landing heads all five times is \(1/32\) or approximately 0.03125.

Step 1: Identify the probability of a single toss

First, we need to determine the probability of the coin landing heads in a single toss. For a fair coin, there are two possible outcomes: heads and tails. Both outcomes have an equal chance of happening, so the probability of getting heads is 1/2 (one out of two possible outcomes).

Step 2: Use the multiplication rule for independent events

Since the coin tosses are independent events, we can use the multiplication rule to find the probability of getting heads all five times. This rule states that the probability of multiple independent events occurring is the product of their individual probabilities. In this case: P(getting heads all 5 times) = P(getting heads in the 1st toss) × P(getting heads in the 2nd toss) × P(getting heads in the 3rd toss) × P(getting heads in the 4th toss) × P(getting heads in the 5th toss)

Step 3: Calculate the probability

Using the probability of getting heads in a single toss (1/2) and applying the multiplication rule, we can now calculate the probability of getting heads all five times: P(getting heads all 5 times) = (1/2) × (1/2) × (1/2) × (1/2) × (1/2) = (1/2)^5 = 1/32 So the probability of the coin landing heads all five times is 1/32 or approximately 0.03125.

Key Concepts

Probability Theory

Understanding probability theory is foundational to analyzing random events and is crucial in various fields such as finance, insurance, and science. In essence, probability quantifies the likelihood of an event occurring on a scale from 0 to 1, where 0 indicates impossibility and 1 denotes certainty.

When flipping a fair coin, as in our example, probability theory allows us to determine that there is an equal chance of landing on heads or tails. This simple scenario can be expanded into more complex situations, where the probability of several events occurring in sequence or at once needs to be calculated. Often, this involves determining whether events affect one another, which leads to the principle of independent events and the use of the multiplication rule to calculate combined probabilities.

Multiplication Rule

The multiplication rule is a fundamental concept within probability theory used to find the likelihood of two or more independent events happening together. When events are independent, the outcome of one event does not influence the outcome of another. For instance, multiple tosses of a fair coin are independent because the result of one toss does not affect the others.

To apply the multiplication rule effectively, make a note of the following steps:

• Confirm that the events are independent.
• Calculate the probability of each event occurring separately.
• Multiply these individual probabilities together.

This rule is the basis for finding the combined probability of independent events and explains why in our exercise, the probability of the coin landing heads five times in a row is simply the product of the probability of landing heads in each individual toss.

Independent Events

Independent events are a key concept in probability that refer to scenarios where the outcome of one event does not impact the outcome of another. For our coin flips, whether we get heads or tails on the first, second, or any subsequent flip is independent of the flips before it. This is a critical point that validates the use of the multiplication rule.

In real-world examples, independence may not always be as clear. For instance, the probability of drawing a red card from a deck of cards would change if one card is removed and not replaced, making subsequent events dependent. Understanding the distinction between independent and dependent events is paramount in accurately calculating probabilities and avoiding errors in statistical analysis.