Problem 59 - Algebra for College Students

Question

What are the odds in favor of getting three heads with a toss of three coins? 1 to 7

Step-by-Step Solution

Answer

1 to 7

Step 1: Determine Total Outcomes

When tossing three coins, each coin has two possible outcomes: heads (H) or tails (T). Therefore, the total number of outcomes is calculated by raising the number of possible outcomes (2) to the power of the number of coins (3): \[ 2^3 = 8. \]So, there are 8 possible outcomes when tossing three coins.

Step 2: Identify Favorable Outcomes

For the event of getting three heads (HHH), we need to count how many ways this can occur. In this case, there is only one outcome that corresponds to all three coins being heads, and that is HHH.

Step 3: Calculate the Odds in Favor

Odds in favor are calculated as the ratio of the number of favorable outcomes to the number of unfavorable outcomes. From Step 1, we found that there are 8 total outcomes. From Step 2, we know there is 1 favorable outcome (HHH). Therefore, the number of unfavorable outcomes is:\[ 8 - 1 = 7. \]The odds in favor are then:\[ \frac{1}{7}. \]

Key Concepts

Coin Toss

A coin toss is one of the most fundamental examples used to teach probability. Each toss of a coin gives you two possible outcomes: heads or tails. In mathematics, this is a classic example of a binary outcome.
When you toss a single coin, you have no influence over whether it lands on heads or tails. This makes each toss independent of any other toss. Thus, the probability of a head in a single toss is independent and always 50% or 0.5.

• Each coin has two possible outcomes.
• Probability for either outcome in a single event is 0.5.
• Multiple tosses mean multiplying the number of possibilities.

For three coins, we use the formula to find all combinations. This is expressed mathematically as \( 2^3 = 8 \) outcomes. This is because each coin could land in one of two ways, and you have three coins.

Odds Calculation

Calculating odds is somewhat different from calculating probability but is closely related. Odds compare the chances of a certain event happening with it not happening.
First, know the total possible outcomes and the number of ways the event can succeed. In the case of three coin tosses, these calculations become more vivid.

• Total possible outcomes: 8 (from three coins)
• Favorable outcomes: 1 (all heads)
• Unfavorable outcomes: 7 (anything but all heads)

Here, odds in favor are calculated as the ratio of favorable outcomes to the unfavorable outcomes. It's written as \( \frac{1}{7} \) for three heads in three tosses. Odds give a clearer picture as to how many times one can expect not to get the event compared to getting it.

Favorable Outcomes

In probability, favorable outcomes are those that match the exact event we are looking for. When calculating likelihoods, it's crucial to count how many "favorable situations" exist amidst all possible combinations.
For example, if your goal is to get three heads when tossing three coins, the favorable outcome is exactly one: HHH.

• Favorable outcome for three heads: 1 (HHH)
• Favorable means the specific outcome that matches your event exactly.

Knowing the favorable outcomes lets you quickly understand the probability or odds of an event. By counting only these specific outcomes, you ignore irrelevant scenarios and focus on what counts for your specific question. This is essential for calculating odds and probabilities effectively.