Problem 61 - Algebra for College Students

Question

What are the odds against getting three heads and two tails with a toss of five coins? 11 to 5

Step-by-Step Solution

Answer

The odds against getting three heads and two tails in five coin tosses are 11 to 5.

Step 1: Understand the Problem

We need to find the odds against getting exactly three heads and two tails when tossing five coins.

Step 2: Calculate Total Possible Outcomes

Each coin has 2 outcomes (head or tail), so with 5 coins, the total number of possible outcomes is given by the formula \(2^5\). Thus, \(2^5 = 32\) total possible outcomes.

Step 3: Calculate Favorable Outcomes

To find the number of favorable outcomes for exactly 3 heads and 2 tails, we use the combination formula. The number of favorable outcomes is given by \(\binom{5}{3}\), because we are choosing 3 coins to be heads out of 5. Calculating this, \(\binom{5}{3} = 10\).

Step 4: Compute Odds Against

Odds against an event are calculated as the ratio of unfavorable outcomes to favorable outcomes. Since there are 10 favorable outcomes, then the number of unfavorable outcomes is \(32 - 10 = 22\). Hence, the odds against are \(22:10\). Simplified, these odds are \(11:5\).

Key Concepts

Coin Toss

A coin toss is a simple probabilistic experiment where a coin is flipped in the air and allowed to land flat on the ground. Each toss results in one of two possible outcomes: heads or tails. Coin tossing is often used as an introductory example in probability theory because of its binary outcome.

When dealing with a series of coin tosses, each toss is independent from the others. This means that the outcome of one toss does not affect the outcome of the next. For example, if you flip a coin five times, the result of each individual flip won't influence the next. However, the total sequence of events can be analyzed for different patterns, such as getting a certain number of heads or tails.

The probability of getting a specific outcome on a single toss is 1/2 or 50%. This simple probability increases in complexity as more coins are tossed because each additional coin doubles the possible outcomes.

Odds Calculation

Odds calculation is a common method to express the likelihood of an event occurring. In probability, we often talk about the chances of something happening in terms of odds. Odds show the ratio of the number of ways an event can occur versus the number of ways it cannot.

To calculate the odds against an event, you first find the total number of possible outcomes and the number of favorable outcomes (ways the event can occur).

• Calculate the total possible outcomes for the experiment.
• Determine the favorable outcomes for the particular event.
• Subtract the number of favorable outcomes from the total to find the unfavorable outcomes.
• Express the odds as the ratio of unfavorable outcomes to favorable outcomes.

For our problem, we calculated the odds against getting three heads and two tails as follows: First, we found the total possible outcomes (32), then the favorable outcomes (10). Subtracting favorable from total outcomes gave us the unfavorable outcomes (22). Thus, the odds against are expressed as a ratio of 22 to 10, simplifying to 11 to 5.

Combinatorics

Combinatorics is a branch of mathematics concerning the counting, arrangement, and combination of objects. It's especially useful in calculating probabilities where multiple outcomes can occur in various arrangements.
In coin-tossing problems like the example above, combinatorics helps us count the number of favorable arrangements of heads and tails. We use the combination formula, \binom{n}{k}\, to determine how many ways we can choose \(k\) successes (e.g., heads) from \(n\) trials (e.g., flips).

• In our exercise, \(n = 5\) because we're flipping 5 coins.
• We want \(k = 3\) heads, so we calculate \binom{5}{3}\, resulting in 10.

This tells us there are 10 ways to get exactly 3 heads in 5 coin flips. Combinatorics simplifies the counting process, allowing us to compute favorable outcomes easily.