Write

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Write

• What is the binomial probability model?
• What do you need to know to use it?
• What are the relevant formulas?
• How do we use binomial probabilities to compute
  for probabilities of less than or greater
  than?

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The Normal Approximation to the Binomial
Distribution

• Wednesday 9 January 2008

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Terms

• p probability of success
• q probability of failure
• of course, q 1 - p

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The Question

• If we perform n Bernoulli trials with a
  probability of success p, what is the probability
  of exactly x successes?

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The Binomial Model!!!

• The probability of X successes if we do N trials
  is equal to
• This is called the binomial model. Bask in its
  all-knowing and all-powerful glow!

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The Binomial Model for Bernoulli Trials

• n is the number of trials.
• x is the number of successes.
• p is the probability of success
• q is the probability of failures
• that (n x) thing is n choose x youve
  probably seen it before looking like nCx

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At Least and At Most

• When faced with an at least situation, remember
  that the probability of a union of events is
  equal to the sum of the probabilities of each
  event.
• So for this problem, the probability that it will
  rain on at least five days is equal to
• the probability that it will rain on five days
  plus
• the probability that it will rain on six days
  plus
• the probability that it will rain on all seven
  days.
• (when the outcomes are exclusive, as they are
  here).

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More Than and Less Than

• There is a 58 chance that the café will serve
  meatloaf on any given day. What is the
  probability that they will server it at least 8
  times in the next 10 days?

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The problem with this approach

• You flip a coin 100 times. What is the
  probability that you flip at least 45 heads?

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A Useful Out

• For any random variable X with a binomial
  distribution, the random variable
• Is normally distributed with a mean of 0 and
  variance of 1.

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WHAT?

• Thats right.
• For any binomial random variable, we can
  approximate for less than using the CDF of the
  normal distribution.

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The Binomial Model

• The binomial model has these values for its mean
  and standard deviation.
• µ np
• s ?npq
• I can show you why, but I wont.
• The mean probably makes sense to you, in any
  event.

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So What?

• We can find the mean and standard deviation, and
  then build a variable thats normally distributed
  from a binomial distribution.
• Then, we can use our normal distribution chart to
  figure out the probability of the given value for
  this new random variable!

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Normal Approximation to the Binomial Model

• Compute the mean for the binomial proability.
• Compute the SD for the binomial probability.
• Normalize the variable for the given value.
• Go to your chart, and get the probability of
  less than that value.
• Where necessary, manipulate this to get an answer.

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Normal Approximation

• You flip a coin 100 times. What is the
  probability that you flip at least 45 heads?

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• What is the likelihood, rolling a die 100 times,
  that you get fewer than 40 even numbers?
• A list of 1000 random 2-digit numbers (10 99)
  contains fewer than 100 numbers over 85. How
  likely is this?
• If you draw a card from a deck 200 times, what is
  the probability of drawing more than 112 red
  cards?
• If it snows one day a week, at random, over 20
  weeks, what is the probability that it snows on
  school days at least 16 times? (ignore vacation
  days)
• You roll a die 300 times. What is the
  probability of fewer than 45 threes?
• Tough One What is the probability, in 500 hands
  of 5-card poker, of being dealt exactly 1 pair at
  least 100 times?

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Normal Approximation to the Binomial Model

• Compute the mean for the binomial proability.
• Compute the SD for the binomial probability.
• Normalize the variable for the given value.
• Go to your chart, and get the probability of
  less than that value.
• Where necessary, manipulate this to get an answer.