Binomial Distribution

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When you flip a coin, there are two possible outcomes: heads and tails.

  • Each outcome has a fixed probability, the same from trial to trial.
  • In the case of coins, heads and tails each have the same probability of 1/2.

More generally, there are situations in which the coin is biased, so that heads and tails have different probabilities.

  • In the present section, we consider probability distributions for which there are just two possible outcomes with fixed probabilities summing to one.
  • These distributions are called binomial distributions.


Example

Outcome First Flip Second Flip
1 Heads Heads
2 Heads Tails
3 Tails Heads
4 Tails Tails

The four possible outcomes that could occur if you flipped a coin twice are listed above in the table.

  • Note that the four outcomes are equally likely: each has probability 1/4.
  • To see this, note that the tosses of the coin are independent (neither affects the other).
  • Hence, the probability of a head on Flip 1 and a head on Flip 2 is the product of P(H) and P(H), which is 1/2 x 1/2 = 1/4.
  • The same calculation applies to the probability of a head on Flip 1 and a tail on Flip 2.
  • Each is 1/2 x 1/2 = 1/4.

The four possible outcomes can be classified in terms of the number of heads that come up.

  • The number could be two (Outcome 1), one (Outcomes 2 and 3) or 0 (Outcome 4).
  • Since two of the outcomes represent the case in which just one head appears in the two tosses, the probability of this event is equal to 1/4 + 1/4 = 1/2.
  • The probabilities of these possibilities are shown in the table and figure below.
Number of Heads Probability
0 1/4
1 1/2
2 1/4

ClipCapIt-140526-171758.PNG

The figure above is a discrete probability distribution

  • It shows the probability for each of the values on the X-axis.
  • Defining a head as a "success," the figure shows the probability of 0, 1, and 2 successes for two trials (flips) for an event that has a probability of 0.5 of being a success on each trial.
  • This makes it an example of a binomial distribution.

The Formula for Binomial Probabilities

The binomial distribution consists of the probabilities of each of the possible numbers of successes on N trials for independent events that each have a probability of π (the Greek letter pi) of occurring.

  • For the coin flip example, N = 2 and π = 0.5.

The formula for the binomial distribution is shown below:

ClipCapIt-140526-172100.PNG

where P(x) is the probability of x successes out of N trials, 
N is the number of trials, and 
π is the probability of success on a given trial. 

Applying this to the coin flip example,

ClipCapIt-140526-172121.PNG

In a spreadsheet
=BINOMDIST(0,2,0.5,false)

Example

If you flip a coin twice, what is the probability of getting one or more heads?

  • Since the probability of getting exactly one head is 0.50 and
  • the probability of getting exactly two heads is 0.25,
  • the probability of getting one or more heads is 0.50 + 0.25 = 0.75.

Now suppose that the coin is biased.

  • The probability of heads is only 0.4.
* What is the probability of getting heads at least once in two tosses? 
  • Substituting into the general formula above, you should obtain the answer .64.

Cumulative Probabilities

We toss a coin 12 times. What is the probability that we get from 0 to 3 heads?

  • The answer is found by computing the probability of exactly 0 heads, exactly 1 head, exactly 2 heads, and exactly 3 heads.
  • The probability of getting from 0 to 3 heads is then the sum of these probabilities.
  • The probabilities are: 0.0002, 0.0029, 0.0161, and 0.0537.
  • The sum of the probabilities is 0.073.
  • The calculation of cumulative binomial probabilities can be quite tedious.
  • Therefore we have provided a binomial calculator to make it easy to calculate these probabilities.


Mean and Standard Deviation of Binomial Distributions

Consider a coin-tossing experiment in which you tossed a coin 12 times and recorded the number of heads. If you performed this experiment over and over again, what would the mean number of heads be?

  • On average, you would expect half the coin tosses to come up heads.
  • Therefore the mean number of heads would be 6.

In general, the mean of a binomial distribution with parameters N (the number of trials) and π (the probability of success on each trial) is:

μ = Nπ
where μ is the mean of the binomial distribution


The variance of the binomial distribution is:

σ2 = Nπ(1-π)
where σ2 is the variance of the binomial distribution


Let's return to the coin-tossing experiment.

  • The coin was tossed 12 times, so N = 12.
  • A coin has a probability of 0.5 of coming up heads.
  • Therefore, π = 0.5.


The mean and variance can therefore be computed as follows:

μ = Nπ = (12)(0.5) = 6

σ2 = Nπ(1-π) = (12)(0.5)(1.0 - 0.5) = 3.0.


Naturally, the standard deviation (σ) is the square root of the variance (σ2).

 ClipCapIt-140526-172508.PNG


Quiz

1 Select all that apply. Which of the following probabilities can be found using the binomial distribution?

The probability that 3 out of 8 tosses of a coin will result in heads
The probability that Susan will beat Shannon in two of their three tennis matches
The probability of rolling at least two 3's and two 4's out of twelve rolls of a die
The probability of getting a full house poker hand
The probability that all 5 of your randomly-chosen group members will have passed the midterm
The probability that a student blindly guessing will get at least 8 out of 10 multiple-choice questions correct

Answer >>

The probability that 3 out of 8 tosses of a coin will result in heads

The probability that Susan will beat Shannon in two of their three tennis matches

The probability that all 5 of your randomly-chosen group members will have passed the midterm

The probability that a student blindly guessing will get at least 8 out of 10 multiple-choice questions correct


A binomial distribution has only two possible outcomes. You can think of them as successes and failures. For the correct answers, the successes are: a flip of heads, a win for Susan, a group member who has passed the midterm, and a correct answer on a multiple-choice question.


2 You flip a fair coin 10 times. What is the probability of getting 8 or more heads?

Answer >>

0.55


You may use the Binomial Calculator (n is 10, p is .5, > or equal to 8). Otherwise add up the probability of getting 8, 9, and 10 heads: .044 + .01 + .001 equal to .055


3 The probability that you will win a certain game is 0.3. If you play the game 20 times, what is the probability that you will win at least 8 times?

Answer >>

0.23

Use the Binomial Calculator (n is 20, p is.3, > or equal to 8). p is .23


4 The probability that you will win a certain game is 0.3. If you play the game 20 times, what is the probability that you will win 3 or fewer times?

Answer >>

0.11

Use the Binomial Calculator (n is 20, p is .3, less than or equal to 3). p is .11


5 The probability that you will win a certain game is 0.3. If you play the game 20 times, what is the probability that you will win 3 or fewer times?

Answer >>

6

M is np is 20 x .3 equals to 6



6 A biased coin has a .6 chance of coming up heads. You flip it 50 times. What is the variance of this distribution?

Answer >>

12

Var is np(1-p) is 50(.6)(1-.6) equal to 12