Exam code:1ST0
Characteristics of the Binomial Distribution
What is a probability distribution?
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A probability distribution is a list of all the possible outcomes of an experiment, along with the probabilities for each outcome
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For example, if the experiment is flipping a fair coin then the distribution can be represented in a table
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x stands for a possible outcome
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P(x) is the probability of that outcome occurring
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x |
heads |
tails |
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P(x) |
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Or if the experiment is rolling a fair dice then the distribution can also be given in a table as
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x |
1 |
2 |
3 |
4 |
5 |
6 |
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P(x) |
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What is a binomial distribution?
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A binomial distribution can describe an experiment where
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Something is repeated a number of times
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For example, flipping a fair coin 10 times
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Each repeat is called a trial
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The number of successes is counted
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For example, counting the number of heads you get in the 10 tosses
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A success doesn’t have to be a good thing
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It’s just the name used for one of the outcomes
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To use a binomial distribution, the following conditions must be met
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The number of trials is fixed
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Each trial has two possible outcomes (‘success’ and ‘failure’)
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The trials are independent
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The probability of success in each trial is constant
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The notation B(n, p) is used to denote a binomial distribution
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n is the number of trials
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p is the probability of success
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Because there are only two possible outcomes
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the probability of failure is 1-p
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The probability of failure is often denoted by q
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where q = 1-p
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The mean of the binomial distribution B(n, p) is np
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This is the expected frequency for the number of successes in n trials
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Examiner Tips and Tricks
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Be sure to learn the conditions for when a binomial distribution is appropriate
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Exam questions often ask about these specifically
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Worked Example
Hannah is a snowboarder who is trying to perform the Poptart trick.
Hannah would like to use a binomial distribution to find the probabilities for how many times she will successfully complete the Poptart trick, out of her next 12 attempts.
(a) Give a reason why the binomial distribution might be suitable in this case.
Consider the list of necessary conditions for using a binomial distribution
Write down one that is definitely met here
Relate it to the context
There are a fixed number of trials (her 12 attempts)
‘There are only two possible outcomes, success (doing the trick) and failure (not doing the trick)’ would also get the mark
(b) Suggest a reason why the binomial distribution may not be suitable in this case.
Consider the list of necessary conditions for using a binomial distribution
Write down one that might not be met here
The trials might not be independent, because she might get better each time from practising the trick
‘The probability of success might not be constant’ (for the same reason) would also get the mark
Hannah successfully performs the Poptart trick 20% of the time.
(c) Assuming that using a binomial distribution is suitable, write down the distribution that Hannah could use in the form B(n, p).
The number of trials, n, is 12
The probability of success, p, is 0.2 (20% as a decimal)
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