Problem 24 - College Algebra with Corequisite Support

Question

For the following exercises, four coins are tossed. Find the probability of tossing exactly two heads or at least two tails.

Step-by-Step Solution

Answer

The probability is \( \frac{11}{16} \).

Step 1: Understand the Problem

We need to find the probability of two specific events in a four-coin toss: tossing exactly 2 heads (and thus 2 tails) or tossing at least 2 tails.

Step 2: Find Total Possible Outcomes

When tossing 4 coins, each coin has 2 possible outcomes (Head or Tail). Therefore, there are \(2^4 = 16\) total possible outcomes for the 4-coin toss.

Step 3: Calculate Probability of Exactly 2 Heads

To get exactly 2 heads, we need to calculate the number of favorable outcomes where 2 of the 4 coin tosses result in heads. This can be calculated using the combination formula \( \binom{n}{k} \), which is \( \binom{4}{2} = 6 \).

Step 4: Calculate Probability of At Least 2 Tails

At least 2 tails means 2, 3, or 4 tails. This includes the scenarios of exactly 2 heads (already calculated), plus 3 heads (1 tail), and 4 heads (0 tails). The number of these outcomes are 6, 4, and 1 respectively, totaling to 11 outcomes.

Step 5: Evaluate Overlap and Combine Probabilities

The probability of exactly 2 heads and at least 2 tails overlaps only in the scenario where there are exactly 2 tails. Since the outcomes are independent, we don't double-count. Add probabilities: exactly 2 heads (6 outcomes) and at least 2 tails (11 outcomes).

Step 6: Solve for Probability

Both events include exactly 2 heads, giving 11 favorable outcomes. Therefore, the probability of tossing exactly 2 heads or at least 2 tails is \( \frac{11}{16} \).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics concerned with counting, arranging, and finding patterns in sets. When dealing with problems involving coin tosses, combinatorics helps us determine the number of possible outcomes. For instance, when tossing one coin, there are 2 outcomes: heads or tails.

In the four-coin toss exercise, each coin being independent means that the outcomes multiply. Hence, for 4 coins, we have a total of \( 2^4 = 16 \) possible outcomes. This calculation uses the concept of the 'combination' where each flip is treated independently.

• Total possible flips: 16 (from \( 2^4 \) )
• Exactly 2 heads: Use combinations to find the favorable outcomes \( \binom{4}{2} = 6 \).

Combinatorics through combinations helps in determining scenarios such as finding exactly 2 heads, which is crucial in calculating probabilities in such random experiments.

Binomial Probability

Binomial probability helps us find the probability of a particular number of successes in a fixed number of trials. When tossing coins, each toss is a trial, and getting a 'head' or a 'tail' is considered a success for that side. For a four-coin toss, we can use binomial probability to determine the likelihood of getting exactly or at least a certain number of heads or tails.

To calculate the probability of exactly 2 heads, we apply the binomial formula, which is \( \binom{n}{k} p^k (1-p)^{n-k} \). In this scenario:

• \( n = 4 \) (number of trials)
• \( k = 2 \) (desired number of heads)
• \( p = 0.5 \) (probability of getting heads on one toss)

The formula becomes \( \binom{4}{2} (0.5)^2 (1-0.5)^{2} = \frac{6}{16} \).
Using binomial probability simplifies understanding of how often a sequence of independent trials leads to a specific result.

Outcome Analysis

Outcome analysis involves examining possible outcomes to determine probabilities of various events. When we talk about coin tosses, we track different results, such as heads and tails combinations. In the four-coin toss problem, outcome analysis helps us find favorable outcomes and understand their arrangements.

We calculate the probability of getting at least 2 tails by adding the probabilities of two tails, three tails, and four tails, which represent distinct outcomes:

• Two tails (2 heads): Already calculated with 6 favorable outcomes.
• Three tails (1 head): Combination outcome counts as 4.
• Four tails (0 heads): Single outcome, as all show tails.

The sum results in 11 possible scenarios where at least 2 tails appear, giving a probability of \( \frac{11}{16} \).
By counting and analyzing these outcomes, we ensure accuracy in our probability calculations, giving us insight into patterns and expectations in random events.