# Calculating Binomial Probability for Coin Tosses with At Least 1 Head

• mtingt
In summary, the probability of obtaining exactly 2 heads in 5 coin tosses, given that at least 1 head appeared, is approximately 0.323. This can be calculated using the conditional probability formula and the binomial distribution.
mtingt
A fair coin is tossed 5 times. What is the probability of obtaining exactly 2 heads if it is known that at least 1 head appeared?

The way I see it, this is a conditional probability problem that involves the binomial distribution. That is, "what is the probability of getting exactly 2 heads given that at least 1 head occurs".

The formula for the probabilty of event A given B, written as P(A|B), is:
P(A|B) = $\frac{P(A\bigcap B)}{P(B)}$.

In this case, P(A) = probability of exactly 2 heads, and P(B) = probability of ≥ 1 head.
Since P(A) fulfills P(B) as well, P(A$\bigcap$B) = P(A)= (10 choose 2)(0.5^5)
= 0.3125.

P(B) is a bit trickier. It's basically the P(1 H) + P(2 H) + P(3 H) + P(4 H) + P(5 H), which can also be written as 1 - P(0 heads) = 1 - (0.5)^5 = 0.96875.

Therefore, P(A|B) = 0.3125 / 0.96875 ≈ 0.323

EDIT - this is basically the full solution to what mathman just said.

thanks pshooter that explained so much better!

## 1. What is binomial probability?

Binomial probability is a mathematical concept that calculates the likelihood of a specific event occurring multiple times in a set number of trials. It is based on the binomial distribution, which is used to model the probability of a certain number of successes in a fixed number of independent trials.

## 2. How is binomial probability calculated?

Binomial probability is calculated using the formula P(x) = nCx * p^x * (1-p)^(n-x), where n is the total number of trials, x is the number of successes, and p is the probability of success in each trial. nCx, or n choose x, represents the number of ways to choose x objects from a set of n objects.

## 3. What types of problems can binomial probability solve?

Binomial probability can be used to solve problems involving a fixed number of independent trials with a binary outcome (success or failure). This can include coin flips, dice rolls, and other similar scenarios.

## 4. What are some real-world applications of binomial probability?

Binomial probability can be applied in various fields, including statistics, economics, and biology. It can be used to analyze the results of clinical trials, predict the likelihood of success in marketing campaigns, and understand the genetics of inherited traits.

## 5. How can I use binomial probability to solve a problem?

To use binomial probability to solve a problem, you will need to identify the values of n, x, and p and plug them into the formula P(x) = nCx * p^x * (1-p)^(n-x). You may also need to use a calculator or statistical software to compute the solution. It is important to clearly define the problem and understand the assumptions of the binomial distribution before applying the formula.

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