Question regarding binomial random variable and distribution

In summary, the binomial random variable X, associated with a binomial experiment of n trials, is defined as the number of S's among the trials. In this case, with n = 3, there are 8 possible outcomes for the experiment, which can be calculated by multiplying the number of possible results for each trial (2) by the number of trials (3). If order does not matter, the formula is different. Additional resources on binomial coefficients and combinations can be found at the provided links.
  • #1
level1
4
0
Hi,

just started learning probability & need some help in understanding...

"The binomial random variable X associated with a binomial experiment consisting of n trials is defined as

X = the number of S's among the n trials.

Suppose, for example, that n = 3. Then there are 8 possible outcomes for the experiment:

SSS SSF SFS SFF FSS FSF FFS FFF"

Why is it that there will be 8 possible outcomes?
 
Physics news on Phys.org
  • #2
You have three trials--each trial has two possible results: S or F. Computing the total number of outcomes of the three trials is a matter of just adding together all of the separate results.

First trial: 2 results, Second trial: 2 results, Third trial: 2 results

2*2*2 = 23 = 8 outcomes

The general formula is: Number of outcomes = pn, where p is the number of possible results of each trial, and n is the number of trials.

If order doesn't matter, then in your example SSF and FSS would be considered the same outcome since the same number of S and F occur in both. The formula for this is different. Here's a link for more, I'm late for class...

http://en.wikipedia.org/wiki/Binomial_coefficient
http://mathworld.wolfram.com/Combination.html
 
  • #3
Thanks very much for the explanation. I understand why there 8 possible outcomes now. thanks again!
 

What is a binomial random variable?

A binomial random variable is a discrete random variable that represents the number of successes in a fixed number of independent trials. It follows a binomial distribution, which is a type of probability distribution that describes the likelihood of different outcomes in a binomial experiment.

What are the characteristics of a binomial random variable?

A binomial random variable has the following characteristics:

  • There are a fixed number of trials, denoted by n.
  • Each trial has only two possible outcomes, typically referred to as success and failure.
  • The probability of success, denoted by p, is the same for each trial.
  • The trials are independent, meaning that the outcome of one trial does not affect the outcome of any other trial.

How is the binomial distribution calculated?

The binomial distribution is calculated using the following formula:

P(X = k) = (n choose k) * pk * (1-p)n-k

Where n is the number of trials, p is the probability of success, and k is the number of successes. (n choose k) is a combination function that calculates the number of ways to choose k successes from n trials.

What is the difference between a binomial random variable and a binomial distribution?

A binomial random variable is a specific outcome of a binomial experiment, while a binomial distribution is a probability distribution that represents all possible outcomes of a binomial experiment. In other words, a binomial random variable is a single value, while a binomial distribution is a set of probabilities for different values.

How is a binomial random variable used in real life?

Binomial random variables are commonly used in real life to model events with two possible outcomes, such as coin flips, success/failure experiments, and survey responses. They are also used in statistics and data analysis to test hypotheses and make predictions about future events.

Similar threads

I Sample space, outcome, event, random variable, probability...
    • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
I Random variable vs Random Process
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
I Random variable and elementary events
    • Set Theory, Logic, Probability, Statistics
Replies
9
Views
551
I Linear regression and random variables
    • Set Theory, Logic, Probability, Statistics
Replies
30
Views
2K
I Probability of White Ball in Box of 120 Balls: Solved!
    • Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
Chernoff Bounds using importance sampling identity
    • Set Theory, Logic, Probability, Statistics
Replies
0
Views
1K
MHB Multiple choice test : random variable
    • Set Theory, Logic, Probability, Statistics
Replies
7
Views
956
B Does the binomial distribution play a role determining p from data?
    • Set Theory, Logic, Probability, Statistics
Replies
4
Views
1K
MHB Binomial Distribution: Likelihood Ratio Test for Equality of Several Proportions
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
2K
B Definition of a random variable in quantum mechanics?
    • Quantum Physics
Replies
12
Views
737

Hot Threads

Recent Insights

Back
Top