Binomial Distribution

Definition

Given random variable :

where is the number of successes in a sequence of independent Bernoulli trials with a probability of success .

Intuition

An alternate formulation in terms of the Bernoulli Distribution:

If are independent and , then .

In fact, the binomial distribution is a generalization of the Bernoulli distribution.

Property

If , , and and are independent, then

Support and PMF

(support and p.m.f.)

Support

PMF

In a binomial experiment of trials, the probability of successes is given by:

where:

  • is the probability of success
  • is the probability of failure

(see choose)

Expected Value and Variance

(expected value and variance)

Definition

Given :

Examples

Example

Suppose that a coin is tossed 10 times with a biased coin where the probability of getting a head is 60%. What is the probability of getting 7 heads?

solution

So

Intuitively, after 10 tosses with 3 tails and 7 heads, there are arrangements. Then we want 7 heads AND 3 tails, so we multiply the probabilities.

Example

There are 6 questions on a multiple-choice quiz. Each has 4 possible answers. Joe has not studied for the quiz, and guesses every question. What is the probability of the following events:

  1. Joe gets all 6 questions correct:
  1. Joe gets at least 2 questions correct:
Example

In a production line of cars, the probability of a car selected at random being flawless is . What is the probability that the sample of ten cars taken off the line will be an even split of 5 flawless cars and 5 cars that need fixing?

solution

Example

Consider the binomial experiment where 5 playing cards are tossed and the number of cards that land face-up () is recorded. The cards are folded and weighted in such a way that the probability of each individual card landing face-up is 0.25.

  1. Write a binomial expression that calculates the probability of any given of any given number of face-up cards from 0 to 5 cards
    Let represent the number of cards that land face up
  1. Find the expected number of cards out of 5 that will land face-up
  1. What is the probability of exactly 3 cards landing face-up?
  1. If the experiment is conducted times, how many times would you expect to see exactly 3 cards land face up?
Example

In a game, the probability that a player wins is about 49%. One evening, Gordon Ramsay decides to play 14 rounds of this game.

  1. What is the probability that Gordon loses 2 times or less?
    Let represent the # of losses
  1. Calculate the expected value for the number of wins that evening
    Let represent the number of wins
Example

Donald Trump picks up 10 cards with replacement from a well-shuffled 52-card deck. If the number of hearts, then:

  1. Find the CDF of at 3
  2. Find
  3. Find

solution

You can't use 'macro parameter character #' in math mode\begin{align} P(1 < X \le 4) & = P(X = 2) + P(X = 3) + P(X = 4) \\ & = \binom{10}{2}\left( \frac{1}{4} \right) { #2} \left( \frac{3}{4} \right) { #8}

{ #3}
\left( \frac{3}{4} \right)

& \approx 0.6778
\end

\begin{align}
E[X] & = np \
& = 10 \left( \frac{1}{4} \right) \
& = 2.5
\end

Example

Nasif Qadri tosses a fair coin times. What is the probability that the first toss is a head given we know he had heads in trials.

solution
= first toss is an H
= heads in coin tosses