Binomial Distributions

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Section 6.3

• Binomial Distributions

2
A Gaggle of Girls

• Lets use simulation to find the probability that
  a couple who has three children has all girls.
• P(girl) 0.5
• Let 0 boy and 1 girl.
• Use your calculator to choose 3 random digits to
  simulate this experiment.
• Complete this experiment 50 times in your group
  and record. Create a probability distribution
  for X number of girls.

3
Gaggle continued

• What was your groups probability for having
  three girls?
• Use your knowledge of probabilities to find the
  actual chance that a family with three children
  has three girls.
• Are these close?

4
Children, Again???

• Two types of scenarios
• A couple is going to have children until they
  have a girl.
• Here, the random variable is how many children
  will it take to get a girl.
• A couple is going to have 3 children and well
  count how many are girls.
• Here, the random variable is how many girls there
  are out of the 3 children.

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Dichotomous Outcomes

• Both of those situations have dichotomous (two)
  outcomes.
• Other examples with two outcomes
• Coin toss (heads or tails)
• Shooting free throws (make or miss)
• A game of baseball (win or lose)

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Special Type of Setting

• In this chapter, well study a setting with two
  outcomes where there are a fixed number of
  observations (or trials).
• The binomial distribution is a special type of
  setting in which there are two outcomes of
  interest.

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4 Conditions for a Binomial Setting

1. There are two outcomes for each observation,
   which we call success or failure.
2. There is a fixed number n of observations.
3. The n observations are all independent.
4. The probability of success, called p, is the same
   for each observation.

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Binomial Random Variables

• Binomial random variable In a binomial setting,
  the random variable X of success.
• The probability distribution of X is called a
  binomial distribution.
• The parameters of a binomial distribution are n
  (the number of observations) and p (the
  probability of success on any one observation).
• B(n, p)
• Is a binomial random variable discrete or
  continuous?

Discrete
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Example

• Blood type is inherited. If both parents have
  the genes for the O and A blood types, then each
  child has probability 0.25 of getting two O genes
  and thus having type O blood. Is the number of O
  blood types among this couples 5 children a
  binomial distribution?
• If so, what are n and p?
• If not, why not?

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Example

• Deal 10 cards from a well-shuffled deck of cards.
  Let X the number of red cards. Is this a
  binomial distribution?
• If so, what are n and p?
• If not, why not?

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Using the Calculator to Find Binomial
Probabilities

• Under 2nd VARS (DISTR), find 0binompdf(
• This command finds probabilities for the binomial
  probability distribution function.
• The parameters for this command are
  binomialpdf(n, p, x) IN THAT ORDER.
• This will only give you the probability of a
  single x value.

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Example

• Lets go back to the couple having three
  children. Let X the number of girls.
• p P(success) P(girl) 0.5
• The possible values for X is 0, 1, 2, 3.
• Using the binompdf(n,p,x) command, complete the
  probability distribution.
• What is the probability that the couple will have
  no more than 1 girl?

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Cumulative Distribution Function

• The pdf command lets you find probabilities for
  ONE value of X at a time.
• binomialcdf(n, p, x)
• This time, you will be given the sum of the
  probabilities x. Be sure you remember this
  when answering a question
• The cdf command finds cumulative probabilities.
  We can use it to quickly find probabilities such
  as P(X lt 7) or P(X 4).

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Corinnes Free Throws

• Corinne makes 75 of her free throws over the
  course of a season. In a key game, she shoots 12
  free throws and makes 7 of them. Is it unusual
  for her to shoot this poorly or worse?
• What is the probability that Corinne makes at
  least 6 of the 12 free throws?

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Homework

• Chapter 6
• 69-72, 86, 94