Problem 11 - Algebra 2

Question

Find each probability if a coin is tossed 5 times. \(P(\text { exactly } 2 \text { tails })\)

Step-by-Step Solution

Answer

The probability is \(\frac{5}{16}\).

Step 1: Understand the Scenario

A coin is tossed 5 times. Each toss results in either heads (H) or tails (T). We want to find the probability of getting exactly 2 tails in these 5 tosses.

Step 2: Identify the Total Number of Outcomes

Since each coin toss has 2 possible outcomes (H or T), and the coin is tossed 5 times, the total number of possible outcomes is calculated as \(2^5 = 32\).

Step 3: Determine the Number of Favorable Outcomes

We need to find the number of ways to get exactly 2 tails in 5 tosses. This can be thought of as choosing 2 positions from 5 for tails, and the remaining will be heads. Use the combination formula \(\binom{n}{k}\), where \(n\) is the total number of trials and \(k\) is the number of successful trials (tails in this case). Here, \(\binom{5}{2} = 10\).

Step 4: Calculate the Probability

The probability of getting exactly 2 tails is the number of favorable outcomes divided by the total number of outcomes. Thus, \[P(\text{exactly 2 tails}) = \frac{\binom{5}{2}}{32} = \frac{10}{32} = \frac{5}{16}\].

Key Concepts

Coin Toss

A coin toss is one of the simplest experiments in probability. When you flip a coin, there are two possible results: you either get heads (H) or tails (T). This makes coin tossing a perfect example of a binary outcome experiment. Each flip of the coin is an independent event, meaning the result of one flip does not affect the result of another. In probability terms, this is called an independent trial.

• The probability of landing a head is 0.5.
• The probability of landing a tail is also 0.5.

When a coin is tossed multiple times, like 5 times, each toss remains independent. The outcomes of multiple tosses can be mapped out to show all possible results, which can be useful for calculating more complex probabilities.

Binomial Distribution

When we are interested in the outcomes of multiple independent binary trials, such as tossing a coin multiple times, we often use a binomial distribution to model the data. This type of distribution is helpful when we want to know the probability of a certain number of successes (like getting tails) in a fixed number of trials.

• The number of trials (n) is fixed.
• Each trial has two possible outcomes, like success (tails) and failure (heads).
• The probability of success is the same for each trial.

In our scenario of tossing a coin 5 times, the binomial distribution can tell us the probability of getting exactly 2 tails. We can compute this using mathematical formulas tied to the distribution principles.

Combination Formula

The combination formula is a powerful tool in probability, especially in binomial distribution problems. It allows us to calculate the number of ways we can choose a subset of items from a larger set, where order does not matter.
For example, in our coin toss problem, we want to know how many ways we can choose exactly 2 tails from 5 tosses. This is represented by the formula \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
where:

• \(n\) is the total number of items (tosses),
• \(k\) is the number of items to choose (tails).

Plugging in the numbers, for \(\binom{5}{2}\), we find there are 10 ways to get exactly 2 tails.

Outcome Calculation

Calculating the probability of a specific outcome involves understanding both the total possible outcomes and the number of successful outcomes. In coin toss experiments, the total number of outcomes after several tosses is calculated with powers of 2. For 5 tosses, it is \(2^5 = 32\) possible outcomes.
To find the probability of getting 2 tails, we divide the number of successful outcomes (as calculated with the combination formula) by the total number of possible outcomes from the coin tosses. For example, using the choices from the combination formula (10 ways for 2 tails), we calculate:

• \[ P(\text{exactly 2 tails}) = \frac{\binom{5}{2}}{32} = \frac{10}{32} = \frac{5}{16} \]

This shows us that the probability of getting exactly 2 tails when tossing a coin 5 times is \(\frac{5}{16}\). Each step of this calculation helps us understand how likelihood and possibilities work together in probability experiments.