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Binomial Distribution

Binomial Distribution. The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial. These trials are often referred to as Bernoulli trials E.g. Flipping a Coin (Head or Tail) Rolling a Dice – HUH ??????

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Binomial Distribution

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  1. Binomial Distribution The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial. These trials are often referred to as Bernoulli trials E.g. Flipping a Coin (Head or Tail) Rolling a Dice – HUH ?????? Well we could say the outcomes are getting a 6 and not getting a 6

  2. These outcomes are appropriately labeled "success" and "failure". E.g. getting a 6 = success The binomial distribution is used to obtain the probability of observing x successes in N trials, with the probability of success on a single trial denoted by p.E.g. for each trial chance of a 6 =The binomial distribution assumes that p is fixed for all trials.

  3. Binomial Experiment • Fixed number of trials = n • Only two outcomes , success (p) or failure (q) • Each trial is independent and identical • Probability, p never changes in any trial and q (failure) = 1 – p and also never changes

  4. WHY? Tree diagrams can only go so far before they become cumbersome. E.g. Rolling a Die in successive trials exciting huh!

  5. 6 3 rolls of a dice 6 6’ 6 6 6’ 6’ 6 6 6’ 6’ Paths in Red show those outcomes which have two sixes 6’ How many paths are there? 6’ What is the probability at the end of each path?

  6. Deriving the rule There are 3 branches Each branch has a probability of WHY are they the same? So the total chance of rolling two sixes in 3 trials is

  7. Let’s look at the problem again using our new variables Let p = rolling a 6 = 1/6 , q = not rolling a 6 =5/6 Let n = trials = 3 And let r = number of successes you desire THE PREVIOUS CALCULATION Would look like No. arrangements of (ppq)

  8. Arrangements of p and q ppq = 3 arrangements from the tree diagram What if you carried out 8 trials and wanted the probability of getting 2 sixes. THAT IS A BIG TREE! How many arrangements of ppqqqqq are there? That’s right it is a combination!

  9. SO NOW TO WRAP IT UP! • The rule is now complete And the good news The Calculator has a function to do it all for you…Binompdf which is in the DISTR Menu Just enter in (n, p, r)

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