Bernard Szlachta: /* The Formula for Binomial Probabilities */

2014-05-28T16:29:15Z

The Formula for Binomial Probabilities

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{{Cat|Probability| 05}}

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=
{| class="wikitable" style="text-align: center;"
|-
! 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.

{| class="wikitable" style="text-align: center;"
|-
! Number of Heads
! Probability
|-
| 0
| 1/4
|-
| 1
| 1/2
|-
| 2
| 1/4
|}
[[File:ClipCapIt-140526-171758.PNG| 400px]]

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:
[[File: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,

[[File: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).
[[File:ClipCapIt-140526-172508.PNG]]

=Quiz=
<quiz display=simple>

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

|type="[]"}
+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

{
{{Show 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.
}}
}

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

|type="{}"}
{ 0.55 | .55 }

{
{{Show 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
}}
}

{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?

|type="{}"}
{ 0.23 | .23 }

{
{{Show Answer|
0.23

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

{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?

|type="{}"}
{ 0.11 | .11 }

{
{{Show Answer|
0.11

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

{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?

|type="{}"}
{ 6 }

{
{{Show Answer|
6

M is np is 20 x .3 equals to 6

}}
}

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

|type="{}"}
{ 12 }

{
{{Show Answer|
12

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

}}
}

</quiz>

Binomial Distribution - Revision history

Bernard Szlachta: /* The Formula for Binomial Probabilities */