The binomial distribution, which is a type of discrete probability distribution, consists of
the probabilities of the outcomes of a binomial experiment, and these experiments come from
probability problems that have only two outcomes, or can be reduced to two outcomes. For instance a
coin flip can be either a head or a tail, and on a test a multiple-choice question can be reduced to
correct or incorrect. There are 4 requirements for a binomial experiment. Number 1 is that there
must be a fixed number of trials. Number 2 is that each trial can have only 2 outcomes or
be reduced to 2 outcomes, and these are often referred to as a success or a failure. Number
3 is that the outcomes of each trial must be independent of one another, and number 4 is that
the probability of a success must be the same for each trial. Binomial probability is the probability
of x successes in n trials. For instance, if you toss a coin 3 times, what is the probability
of getting exactly 2 heads? Checking this problem against the binomial requirements,
there are a fixed number of trials, in this case 3 trials, there only 2 outcomes for each
trial, heads or tails, and we could term getting a head as a success, and a tail as a failure. The
outcome of each trial is independent as no toss has any effect on another toss, and the probability
of success, getting a head, is the same for each trial, there is a 50% probability of getting a
head on each trial. There are two main ways to find the probability of our question, the first is to
use a tree diagram and the addition rule, as you see here. And the probability equals 3 over 8 or
0.375. The second way to find the probability is to use the binomial probability formula. The formula
for getting exactly x successes in n trials is probability of x equals, n factorial divided by n
minus x factorial, times x factorial, times P to the x, times q to the n minus x. This formula may look
a bit intimidating but it's actually not that bad if you break it down, and we'll use our probability
question to do that. So probability of x is the probability of x successes, in our problem this
is the probability of getting 2 heads in 3 tosses, so P of x equals 2 heads. In the formula
n equals the number of trials, and in our problem there 3 trials, so n equals 3. In the formula x
equals the number of successes, and in the problem we're looking for 2 successes, so x equals 2. In
the formula p equals the probability of success in an individual trial, and in our problem the
probability of a success, getting ahead on the flip, is 50% or 1 over 2. And in the formula q equals the
probability of a failure in a single trial, which is 1 minus p, which equals 50% or 1 over 2. Now we
have all that we need, so we plug into the formula, probability of 2 heads equals, 3 factorial, divided
by 3 minus 2 factorial, times 2 factorial, times 1 over 2 squared, times 1 over 2 to the 3 minus 2. And
calculating this out we get 3 over 8 or 0.375, same as the tree diagram. As an example let's say you
took a poll and found that 80% of people check their phone during their workout, if you select
10 people at random, what is the probability that exactly 5 of them check their phone during their
workout? So, first we need to check that this fits the requirements for the binomial experiment. There
are a fixed number of trials, in this example 10, it does consist of just two outcomes, check
their phone or not check their phone, each of the 10 trials is independent of one another, and yes
the probability for success is the same for each trial. In this case, checking their phone during a
workout is a success with a probability of 80% or 0.80. Now that we know it meets the requirements we
can use the formula. So first we need to assign all of our notations for the formula, we need to
assign n, x, p and q. n equals the number of trials, and in the question we're looking at 10 trials, x
equals the number of successes in the n, or 10 trials, and that number is 5, which is given in
the question. p is the probability of success and that is 80% or 0.80, and q is the probability of
failure, which is 1 minus p, which equals 20% or 0.20. And now we're ready to plug and calculate.
So we have probability of x, or probability of 5 successes equals, 10 factorial, divided
by 10 minus 5 factorial, times 5 factorial, times 0.80 raised to the 5th, times 0.20 raised to the 10 minus 5.
I have written a calculation out for this on the screen and the result is
0.0264 or 2.6%, so there is a 2.6% probability of
randomly selecting exactly 5 people out of 10 that check their phone during their
workout. And we can reword the question to get a different perspective. Using the same data, 80%
check their phone, if you randomly select 10 people, what is the probability that at least
7 of them check their phone during their workout. In this question at least 7 means
7, 8, 9 or 10. So in this case, we need to calculate the individual probabilities and
add them up to get the total probability. In using the formula in each of these cases, n will be 10,
p will be 0.80, q will be 0.20, and x will change values,
7, 8 , 9, 10. I have the calculations for each of these cases on the screen. Adding these
up, we get 0.879 or 87.9%. As a note, here are all
the possible values for x, 1 through 10, along with their probabilities. If we add these up, we
get 0.99974, if I had not rounded them off the result would
have been 1, so this checks out because the sum of the probabilities is equal to 1. We
can also compute the mean, variance, and standard deviation for a binomial distribution and these
formulas are listed here. To find the mean we need to know two things, n, the number of trials, and p,
the probability of success. Using our same data, 80% of people check their
phone during a workout, if we randomly selected 1000 individuals, what would the mean be?
So mu, the mean, equals n, in this case 1000, times p, 0.80, which equals
800. And the variance would be Sigma squared equals n, 1000 times p, 0.80,
times q, which again is 1 minus p, or 0.20, and this computes to 160. And to
find the standard deviation, we take the square root of the variance, square root of 160, which
equals 12.65. Alright my friends, that be the basics on the binomial distribution.
Hopefully this video helped you out, I do have more videos right there for you, till next time, I am outta here.